NCERT Solutions for Class 12 Maths Chapter 7 Integrals

NCERT Solutions for Class 12 Maths Chapter 7 Integrals have been designed by subject experts at BYJU’S to help the students in their exam preparations. The Class 12 NCERT Maths Book contains the concept of integrals in chapter 7. In this chapter, students learn about integral calculus (definite and indefinite), their properties and much more. This topic is extremely important for both CBSE board exam and for competitive exams.

The concepts of integrals are given in this chapter in a detailed and easy to understand manner. These Class 12 NCERT Solutions for integrals are very simple and can help the students in understanding the problem solving method very easily. Students can reach for these NCERT Solutions and download it for free to practice them offline as well. These solved questions make the students understand the method of applying the concepts of Integrals in each problem.

Download PDF of NCERT Solutions for Class 12 Maths Chapter 7- Integrals

NCERT Solutions for Class 12 Maths Chapter 7 Integrals
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Access NCERT Solutions for Class 12 Maths Chapter 7- Integrals

Exercise 7.1 page no: 299

Find an anti-derivative (or integral) of the following functions by the method of inspection.

1. sin 2x

2. cos 3x

3. e2x

4. (ax + b)2

5. sin 2x – 4 e3x

Solution:

1. sin 2x

The anti-derivative of sin 2x is a function of x whose derivative is sin 2x

We know that,

NCERT Solutions Mathematics Class 12 Chapter 7 - 1

2. cos 3x

The anti-derivative of cos 3x is a function of x whose derivative is cos 3x

We know that,

NCERT Solutions Mathematics Class 12 Chapter 7 - 2

3. e2x

The anti-derivative of e2x is the function of x whose derivative is e2x

We know that,

NCERT Solutions Mathematics Class 12 Chapter 7 - 3

4. (ax + b)2

The anti-derivative of (ax + b) 2 is the function of x whose derivative is (ax + b)2

NCERT Solutions Mathematics Class 12 Chapter 7 - 4

5. sin 2x – 4 e3x

The anti-derivative of (sin 2x – 4 e3x) is the function of x whose derivative of (sin 2x – 4e3x)

NCERT Solutions Mathematics Class 12 Chapter 7 - 5

Find the following integrals in Exercises 6 to 20:

6.
NCERT Solutions Mathematics Class 12 Chapter 7 - 6

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 7

7.
NCERT Solutions Mathematics Class 12 Chapter 7 - 8

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 9

8.
NCERT Solutions Mathematics Class 12 Chapter 7 - 10

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 11

9.
NCERT Solutions Mathematics Class 12 Chapter 7 - 12

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 13

10.
NCERT Solutions Mathematics Class 12 Chapter 7 - 14

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 15

11.
NCERT Solutions Mathematics Class 12 Chapter 7 - 16

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 17

12.
NCERT Solutions Mathematics Class 12 Chapter 7 - 18

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 19

13.
NCERT Solutions Mathematics Class 12 Chapter 7 - 20

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 21

14.
NCERT Solutions Mathematics Class 12 Chapter 7 - 22

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 23

15.
NCERT Solutions Mathematics Class 12 Chapter 7 - 24

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 25

16.
NCERT Solutions Mathematics Class 12 Chapter 7 - 26

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 27

17.
NCERT Solutions Mathematics Class 12 Chapter 7 - 28

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 29

18.
NCERT Solutions Mathematics Class 12 Chapter 7 - 30

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 31

19.
NCERT Solutions Mathematics Class 12 Chapter 7 - 32

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 33

NCERT Solutions Mathematics Class 12 Chapter 7 - 34

20.
NCERT Solutions Mathematics Class 12 Chapter 7 - 35

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 36

Choose the correct answer in Exercises 21 and 22

21. The anti-derivative of
NCERT Solutions Mathematics Class 12 Chapter 7 - 37equals

(A) (1 / 3) x1 / 3 + (2) x1 / 2 + C

(B) (2 / 3) x2 / 3 + (1 / 2) x2 + C

(C) (2 / 3) x3 / 2 + (2) x1 / 2 + C

(D) (3 / 2) x3 / 2 + (1 / 2) x1 / 2 + C

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 38

NCERT Solutions Mathematics Class 12 Chapter 7 - 39

NCERT Solutions Mathematics Class 12 Chapter 7 - 40

22. If d / dx f (x) = 4x3 – 3 / x4 such that f (2) = 0. Then f (x) is

(A) x4 + 1 / x3 – 129 / 8

(B) x3 + 1 / x4 + 129 / 8

(C) x4 + 1 / x3 + 129 / 8

(D) x3 + 1 / x4 – 129 / 8

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 41

NCERT Solutions Mathematics Class 12 Chapter 7 - 42

NCERT Solutions Mathematics Class 12 Chapter 7 - 43


Exercise 7.2 page no: 304

Integrate the functions in Exercises 1 to 37:

1. 2x / 1 + x2

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 44

2. (log x)2 / x

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 45

NCERT Solutions Mathematics Class 12 Chapter 7 - 46

NCERT Solutions Mathematics Class 12 Chapter 7 - 47

3. 1 / (x + x log x)

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 48

NCERT Solutions Mathematics Class 12 Chapter 7 - 48

4. sin x sin (cos x)

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 49

NCERT Solutions Mathematics Class 12 Chapter 7 - 50

5. Sin (ax + b) cos (ax + b)

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 51

NCERT Solutions Mathematics Class 12 Chapter 7 - 52

NCERT Solutions Mathematics Class 12 Chapter 7 - 53

6. √ax + b

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 54

7. x √x + 2

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 55

NCERT Solutions Mathematics Class 12 Chapter 7 - 56

NCERT Solutions Mathematics Class 12 Chapter 7 - 57

NCERT Solutions Mathematics Class 12 Chapter 7 - 58

8. x √1 + 2x2

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 59

9. (4x + 2) √x2 + x + 1

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 60

10. 1 / (x – √x)

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 61

NCERT Solutions Mathematics Class 12 Chapter 7 - 62

NCERT Solutions Mathematics Class 12 Chapter 7 - 63

NCERT Solutions Mathematics Class 12 Chapter 7 - 64

11. x / (√x + 4), x > 0

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 65

NCERT Solutions Mathematics Class 12 Chapter 7 - 66

NCERT Solutions Mathematics Class 12 Chapter 7 - 67

12. (x3 – 1)1 / 3 x5

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 68

NCERT Solutions Mathematics Class 12 Chapter 7 - 69

13. x 2 / (2 + 3x3)3

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 70

NCERT Solutions Mathematics Class 12 Chapter 7 - 71

14. 1 / x (log x) m, x > 0, m ≠ 1

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 72

15. x / (9 – 4x2)

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 73

NCERT Solutions Mathematics Class 12 Chapter 7 - 74

16. e2x + 3

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 75

17.
NCERT Solutions Mathematics Class 12 Chapter 7 - 76

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 77

NCERT Solutions Mathematics Class 12 Chapter 7 - 78

18.
NCERT Solutions Mathematics Class 12 Chapter 7 - 79

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 80

19.
NCERT Solutions Mathematics Class 12 Chapter 7 - 81

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 82

NCERT Solutions Mathematics Class 12 Chapter 7 - 83

20.
NCERT Solutions Mathematics Class 12 Chapter 7 - 84

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 85

NCERT Solutions Mathematics Class 12 Chapter 7 - 86

NCERT Solutions Mathematics Class 12 Chapter 7 - 87

21.

NCERT Solutions Mathematics Class 12 Chapter 7 - 88

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 89

NCERT Solutions Mathematics Class 12 Chapter 7 - 90

22.

NCERT Solutions Mathematics Class 12 Chapter 7 - 91

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 92

NCERT Solutions Mathematics Class 12 Chapter 7 - 93

23.

NCERT Solutions Mathematics Class 12 Chapter 7 - 94

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 95

24.

NCERT Solutions Mathematics Class 12 Chapter 7 - 96

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 97

25.
NCERT Solutions Mathematics Class 12 Chapter 7 - 97

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 98

NCERT Solutions Mathematics Class 12 Chapter 7 - 99

NCERT Solutions Mathematics Class 12 Chapter 7 - 100

NCERT Solutions Mathematics Class 12 Chapter 7 - 101

26.
NCERT Solutions Mathematics Class 12 Chapter 7 - 102

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 103

27.
NCERT Solutions Mathematics Class 12 Chapter 7 - 104

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 105

NCERT Solutions Mathematics Class 12 Chapter 7 - 106

28.
NCERT Solutions Mathematics Class 12 Chapter 7 - 107

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 108

29.
NCERT Solutions Mathematics Class 12 Chapter 7 - 109

Solution:

Take

log sin x = t

By differentiation we get

NCERT Solutions Mathematics Class 12 Chapter 7 - 110

So we get

cot x dx = dt

Integrating both sides

NCERT Solutions Mathematics Class 12 Chapter 7 - 111

30.

NCERT Solutions Mathematics Class 12 Chapter 7 - 112

Solution:

Take 1 + cos x = t

By differentiation

– sin x dx = dt

By integrating both sides

NCERT Solutions Mathematics Class 12 Chapter 7 - 113

So we get

= – log |t| + C

Substituting the value of t

= – log |1 + cos x| + C

31.

NCERT Solutions Mathematics Class 12 Chapter 7 - 114

Solution:

Take 1 + cos x = t

By differentiation

– sin x dx = dt

NCERT Solutions Mathematics Class 12 Chapter 7 - 115

32.

NCERT Solutions Mathematics Class 12 Chapter 7 - 116

Solution:

It is given that

NCERT Solutions Mathematics Class 12 Chapter 7 - 117

NCERT Solutions Mathematics Class 12 Chapter 7 - 118

33.

NCERT Solutions Mathematics Class 12 Chapter 7 - 119

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 120

NCERT Solutions Mathematics Class 12 Chapter 7 - 121

NCERT Solutions Mathematics Class 12 Chapter 7 - 122

34.

NCERT Solutions Mathematics Class 12 Chapter 7 - 123

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 124

35.

NCERT Solutions Mathematics Class 12 Chapter 7 - 125

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 126

36.

NCERT Solutions Mathematics Class 12 Chapter 7 - 127

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 128

NCERT Solutions Mathematics Class 12 Chapter 7 - 129

37.

NCERT Solutions Mathematics Class 12 Chapter 7 - 130

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 131

NCERT Solutions Mathematics Class 12 Chapter 7 - 132

Choose the correct answer in Exercises 38 and 39.

NCERT Solutions Mathematics Class 12 Chapter 7 - 133

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 134

NCERT Solutions Mathematics Class 12 Chapter 7 - 135

(A) tan x + cot x + C

(B) tan x – cot x + C

(C) tan x cot x + C

(D) tan x – cot 2x + C

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 136

NCERT Solutions Mathematics Class 12 Chapter 7 - 137


Exercise 7.3 Page: 307

1. sin2 (2x + 5)

Solution:-

NCERT Solutions Mathematics Class 12 Chapter 7 - 138

2. sin 3x cos 4x

Solution:-

NCERT Solutions Mathematics Class 12 Chapter 7 - 139

NCERT Solutions Mathematics Class 12 Chapter 7 - 140

3. cos 2x cos 4x cos 6x

Solution:-

By standard trigonometric identity cosA cosB = ½ {cos(A + B) + cos(A – B)}

NCERT Solutions Mathematics Class 12 Chapter 7 - 141

NCERT Solutions Mathematics Class 12 Chapter 7 - 142

4. sin3 (2x + 1)

Solution:-

NCERT Solutions Mathematics Class 12 Chapter 7 - 143

NCERT Solutions Mathematics Class 12 Chapter 7 - 144

5. sin3 x cos3 x

Solution:-

NCERT Solutions Mathematics Class 12 Chapter 7 - 145

NCERT Solutions Mathematics Class 12 Chapter 7 - 146

6. sin x sin 2x sin 3x

Solution:-

NCERT Solutions Mathematics Class 12 Chapter 7 - 147

NCERT Solutions Mathematics Class 12 Chapter 7 - 148

7. sin 4x sin 8x

Solution:-

NCERT Solutions Mathematics Class 12 Chapter 7 - 149

NCERT Solutions Mathematics Class 12 Chapter 7 - 150

Solution:-

NCERT Solutions Mathematics Class 12 Chapter 7 - 151

NCERT Solutions Mathematics Class 12 Chapter 7 - 152

NCERT Solutions Mathematics Class 12 Chapter 7 - 153

Solution:-

NCERT Solutions Mathematics Class 12 Chapter 7 - 154

NCERT Solutions Mathematics Class 12 Chapter 7 - 155

10. sin4 x

Solution:-

NCERT Solutions Mathematics Class 12 Chapter 7 - 156

On simplifying, we get,

NCERT Solutions Mathematics Class 12 Chapter 7 - 157

11. cos4 2x

Solution:-

NCERT Solutions Mathematics Class 12 Chapter 7 - 158

NCERT Solutions Mathematics Class 12 Chapter 7 - 159

NCERT Solutions Mathematics Class 12 Chapter 7 - 160

Solution:-

NCERT Solutions Mathematics Class 12 Chapter 7 - 161

NCERT Solutions Mathematics Class 12 Chapter 7 - 162

NCERT Solutions Mathematics Class 12 Chapter 7 - 163

Solution:-

NCERT Solutions Mathematics Class 12 Chapter 7 - 164

NCERT Solutions Mathematics Class 12 Chapter 7 - 165

NCERT Solutions Mathematics Class 12 Chapter 7 - 166

Solution:-

NCERT Solutions Mathematics Class 12 Chapter 7 - 167

NCERT Solutions Mathematics Class 12 Chapter 7 - 168

15. tan3 2x sec 2x

Solution:-

By splitting the given function, we have,

tan32x sec2x = tan22x tan2x sec2x

From the standard trigonometric identity, tan2 2x = sec2 2x – 1,

= (sec22x -1) tan2x sec2x

By multiplying, we get,

= (sec22x × tan2xsec2x) – (tan2xsec2x)

Integrating both sides,

NCERT Solutions Mathematics Class 12 Chapter 7 - 169

NCERT Solutions Mathematics Class 12 Chapter 7 - 170

16. tan4 x

Solution:-

By splitting the given function, we have,

tan4x = tan2x × tan2x

Then,

From trigonometric identity, tan2 x = sec2 x – 1

= (sec2x -1) tan2x

By multiplying, we get,

= sec2x tan2x – tan2x

Again by using trigonometric identity, tan2 x = sec2 x – 1

= sec2x tan2x- (sec2x-1)

= sec2x tan2x- sec2x+1

Now, integrating on both sides we get,

NCERT Solutions Mathematics Class 12 Chapter 7 - 171

NCERT Solutions Mathematics Class 12 Chapter 7 - 172

Solution:-

NCERT Solutions Mathematics Class 12 Chapter 7 - 173

NCERT Solutions Mathematics Class 12 Chapter 7 - 174

Solution:-

NCERT Solutions Mathematics Class 12 Chapter 7 - 175

NCERT Solutions Mathematics Class 12 Chapter 7 - 176

NCERT Solutions Mathematics Class 12 Chapter 7 - 177

Solution:-

NCERT Solutions Mathematics Class 12 Chapter 7 - 178

NCERT Solutions Mathematics Class 12 Chapter 7 - 179

NCERT Solutions Mathematics Class 12 Chapter 7 - 180

Solution:-

NCERT Solutions Mathematics Class 12 Chapter 7 - 181

NCERT Solutions Mathematics Class 12 Chapter 7 - 182

21. sin-1 (cos x)

Solution:-

Given, sin-1(cosx)

Let us assume cosx = t … [equation (i)]

Then, substitute ‘t’ in place of cosx

NCERT Solutions Mathematics Class 12 Chapter 7 - 183

NCERT Solutions Mathematics Class 12 Chapter 7 - 184

NCERT Solutions Mathematics Class 12 Chapter 7 - 185

Solution:-

NCERT Solutions Mathematics Class 12 Chapter 7 - 186

NCERT Solutions Mathematics Class 12 Chapter 7 - 187

Choose the correct answer in Exercises 23 and 24.

NCERT Solutions Mathematics Class 12 Chapter 7 - 188

Solution:-

NCERT Solutions Mathematics Class 12 Chapter 7 - 189

NCERT Solutions Mathematics Class 12 Chapter 7 - 190

NCERT Solutions Mathematics Class 12 Chapter 7 - 191

Solution:-

NCERT Solutions Mathematics Class 12 Chapter 7 - 192

Let us assume that, (xex) = t

Differentiating both sides we get,

((ex × x) + (ex × 1)) dx = dt

ex (x + 1) = dt

Applying integrals,

NCERT Solutions Mathematics Class 12 Chapter 7 - 193


Exercise 7.4 Page: 315

Integrate the functions in Exercises 1 to 23.

NCERT Solutions Mathematics Class 12 Chapter 7 - 194

Solution:-

NCERT Solutions Mathematics Class 12 Chapter 7 - 195

2.

NCERT Solutions Mathematics Class 12 Chapter 7 - 196

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 197

Substituting the value of t

NCERT Solutions Mathematics Class 12 Chapter 7 - 198

3.

NCERT Solutions Mathematics Class 12 Chapter 7 - 199

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 200

4.

NCERT Solutions Mathematics Class 12 Chapter 7 - 201

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 202

NCERT Solutions Mathematics Class 12 Chapter 7 - 203

5.

NCERT Solutions Mathematics Class 12 Chapter 7 - 204

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 205

6.

NCERT Solutions Mathematics Class 12 Chapter 7 - 206

Solution:

Take x3 = t

We get 3 x2 dx = dt

Integrating both sides

NCERT Solutions Mathematics Class 12 Chapter 7 - 207

On further calculation

NCERT Solutions Mathematics Class 12 Chapter 7 - 208

NCERT Solutions Mathematics Class 12 Chapter 7 - 209

7.

NCERT Solutions Mathematics Class 12 Chapter 7 - 210

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 211

NCERT Solutions Mathematics Class 12 Chapter 7 - 212

NCERT Solutions Mathematics Class 12 Chapter 7 - 213

8.

NCERT Solutions Mathematics Class 12 Chapter 7 - 214

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 215

9.

NCERT Solutions Mathematics Class 12 Chapter 7 - 216

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 217

10.

NCERT Solutions Mathematics Class 12 Chapter 7 - 218

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 219

11.

NCERT Solutions Mathematics Class 12 Chapter 7 - 220

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 221

NCERT Solutions Mathematics Class 12 Chapter 7 - 222

12.

NCERT Solutions Mathematics Class 12 Chapter 7 - 223

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 224

NCERT Solutions Mathematics Class 12 Chapter 7 - 225

Substituting the value of t

NCERT Solutions Mathematics Class 12 Chapter 7 - 226

13.

NCERT Solutions Mathematics Class 12 Chapter 7 - 227

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 228

NCERT Solutions Mathematics Class 12 Chapter 7 - 229

NCERT Solutions Mathematics Class 12 Chapter 7 - 230

NCERT Solutions Mathematics Class 12 Chapter 7 - 231

14.

NCERT Solutions Mathematics Class 12 Chapter 7 - 232

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 233

NCERT Solutions Mathematics Class 12 Chapter 7 - 234

NCERT Solutions Mathematics Class 12 Chapter 7 - 235

15.

NCERT Solutions Mathematics Class 12 Chapter 7 - 236

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 237

NCERT Solutions Mathematics Class 12 Chapter 7 - 238

NCERT Solutions Mathematics Class 12 Chapter 7 - 239

16.

NCERT Solutions Mathematics Class 12 Chapter 7 - 240

Solution:

Consider

4x + 1 = A d/dx (2x2 + x – 3) + B

So we get

4x + 1 = A (4x + 1) + B

On further calculation

4x + 1 = 4 Ax + A + B

By equating the coefficients of x and constant term on both sides

4A = 4

A = 1

A + B = 1

B = 0

Take 2x2 + x – 3 = t

By differentiation

(4x + 1) dx = dt

Integrating both sides

NCERT Solutions Mathematics Class 12 Chapter 7 - 241

We get

= 2 √t + C

Substituting the value of t

NCERT Solutions Mathematics Class 12 Chapter 7 - 242

17.

NCERT Solutions Mathematics Class 12 Chapter 7 - 243

Solution:

Consider

NCERT Solutions Mathematics Class 12 Chapter 7 - 244

It can be written as

x + 2 = A (2x) + B

Now equating the coefficients of x and constant term on both sides

2A = 1

A = ½

B = 2

Using equation (1) we get

NCERT Solutions Mathematics Class 12 Chapter 7 - 245

NCERT Solutions Mathematics Class 12 Chapter 7 - 246

NCERT Solutions Mathematics Class 12 Chapter 7 - 247

18.

NCERT Solutions Mathematics Class 12 Chapter 7 - 248

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 249

NCERT Solutions Mathematics Class 12 Chapter 7 - 250

NCERT Solutions Mathematics Class 12 Chapter 7 - 251

NCERT Solutions Mathematics Class 12 Chapter 7 - 252

NCERT Solutions Mathematics Class 12 Chapter 7 - 253

19.

NCERT Solutions Mathematics Class 12 Chapter 7 - 254

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 255

NCERT Solutions Mathematics Class 12 Chapter 7 - 256

NCERT Solutions Mathematics Class 12 Chapter 7 - 257

NCERT Solutions Mathematics Class 12 Chapter 7 - 258

NCERT Solutions Mathematics Class 12 Chapter 7 - 259

20.

NCERT Solutions Mathematics Class 12 Chapter 7 - 260

Solution:

Consider

NCERT Solutions Mathematics Class 12 Chapter 7 - 261

It can be written as

x + 2 = A (4 – 2x) + B

Now equating the coefficients of x and constant term on both sides

-2A = 1

A = -1/2

4A + B = 2

B = 4

Using equation (1) we get

NCERT Solutions Mathematics Class 12 Chapter 7 - 262

NCERT Solutions Mathematics Class 12 Chapter 7 - 263

NCERT Solutions Mathematics Class 12 Chapter 7 - 264

21.

NCERT Solutions Mathematics Class 12 Chapter 7 - 265

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 266

NCERT Solutions Mathematics Class 12 Chapter 7 - 267

NCERT Solutions Mathematics Class 12 Chapter 7 - 268

NCERT Solutions Mathematics Class 12 Chapter 7 - 269

22.

NCERT Solutions Mathematics Class 12 Chapter 7 - 270

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 271

NCERT Solutions Mathematics Class 12 Chapter 7 - 272

NCERT Solutions Mathematics Class 12 Chapter 7 - 273

NCERT Solutions Mathematics Class 12 Chapter 7 - 274

NCERT Solutions Mathematics Class 12 Chapter 7 - 275

23.

NCERT Solutions Mathematics Class 12 Chapter 7 - 276

Solution:

Consider

NCERT Solutions Mathematics Class 12 Chapter 7 - 277

It can be written as

5x + 3 = A (2x + 4) + B

Now equating the coefficients of x and constant term on both sides

2A = 5

A = 5/2

4A + B = 3

B = -7

Using equation (1) we get

NCERT Solutions Mathematics Class 12 Chapter 7 - 278

NCERT Solutions Mathematics Class 12 Chapter 7 - 279

NCERT Solutions Mathematics Class 12 Chapter 7 - 280

Choose the correct answer in Exercises 24 and 25.

24.

NCERT Solutions Mathematics Class 12 Chapter 7 - 281

(A) x tan -1 (x + 1) + C (B) tan -1 (x + 1) + C

(C) (x + 1) tan -1 x + C (D) tan -1 x + C

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 282

25.

NCERT Solutions Mathematics Class 12 Chapter 7 - 283

NCERT Solutions Mathematics Class 12 Chapter 7 - 284

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 285

NCERT Solutions Mathematics Class 12 Chapter 7 - 286

NCERT Solutions Mathematics Class 12 Chapter 7 - 287


Exercise 7.5 page: 322

Integrate the rational functions in Exercises 1 to 21.

1.

NCERT Solutions Mathematics Class 12 Chapter 7 - 288

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 289

NCERT Solutions Mathematics Class 12 Chapter 7 - 290

2.

NCERT Solutions Mathematics Class 12 Chapter 7 - 291

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 292

NCERT Solutions Mathematics Class 12 Chapter 7 - 293

3.

NCERT Solutions Mathematics Class 12 Chapter 7 - 294

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 295

NCERT Solutions Mathematics Class 12 Chapter 7 - 296

4.

NCERT Solutions Mathematics Class 12 Chapter 7 - 297

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 298

5.

NCERT Solutions Mathematics Class 12 Chapter 7 - 299

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 300

6.

NCERT Solutions Mathematics Class 12 Chapter 7 - 301

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 302

NCERT Solutions Mathematics Class 12 Chapter 7 - 303

7.

NCERT Solutions Mathematics Class 12 Chapter 7 - 304

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 305

NCERT Solutions Mathematics Class 12 Chapter 7 - 306

8.

NCERT Solutions Mathematics Class 12 Chapter 7 - 307

Solution:

We know that

NCERT Solutions Mathematics Class 12 Chapter 7 - 308

It can be written as

x = A (x – 1) (x + 2) + B (x + 2) + C (x – 1)2

Taking x = 1 we get

B = 1/3

Now by equating the coefficients of x2 and constant terms we get

A + C = 0

-2A + 2B + C = 0

By solving the equations

A = 2/9 and C = -2/9

We get

NCERT Solutions Mathematics Class 12 Chapter 7 - 309

NCERT Solutions Mathematics Class 12 Chapter 7 - 310

9.

NCERT Solutions Mathematics Class 12 Chapter 7 - 311

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 312

By further calculation

NCERT Solutions Mathematics Class 12 Chapter 7 - 313

10.

NCERT Solutions Mathematics Class 12 Chapter 7 - 314

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 315

11.

NCERT Solutions Mathematics Class 12 Chapter 7 - 316

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 317

NCERT Solutions Mathematics Class 12 Chapter 7 - 318

12.

NCERT Solutions Mathematics Class 12 Chapter 7 - 319

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 320

13.

NCERT Solutions Mathematics Class 12 Chapter 7 - 321

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 322

14.

NCERT Solutions Mathematics Class 12 Chapter 7 - 323

Solution:

We know that

NCERT Solutions Mathematics Class 12 Chapter 7 - 324

It can be written as

3x – 1 = A (x + 2) + B ….. (1)

Now by equating the coefficient of x and constant terms

A = 3

2A + B = – 1

Solving the equations

B = – 7

We get

NCERT Solutions Mathematics Class 12 Chapter 7 - 325

15.

NCERT Solutions Mathematics Class 12 Chapter 7 - 326

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 327

NCERT Solutions Mathematics Class 12 Chapter 7 - 328

16.

NCERT Solutions Mathematics Class 12 Chapter 7 - 329

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 330

NCERT Solutions Mathematics Class 12 Chapter 7 - 331

17.

NCERT Solutions Mathematics Class 12 Chapter 7 - 332

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 333

NCERT Solutions Mathematics Class 12 Chapter 7 - 334

18.

NCERT Solutions Mathematics Class 12 Chapter 7 - 335

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 336

NCERT Solutions Mathematics Class 12 Chapter 7 - 337

19.

NCERT Solutions Mathematics Class 12 Chapter 7 - 338

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 339

NCERT Solutions Mathematics Class 12 Chapter 7 - 340

20.

NCERT Solutions Mathematics Class 12 Chapter 7 - 341

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 342

NCERT Solutions Mathematics Class 12 Chapter 7 - 343

21.

NCERT Solutions Mathematics Class 12 Chapter 7 - 344

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 345

NCERT Solutions Mathematics Class 12 Chapter 7 - 346

Choose the correct answer in each of the Exercises 22 and 23.

NCERT Solutions Mathematics Class 12 Chapter 7 - 347

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 348

NCERT Solutions Mathematics Class 12 Chapter 7 - 349

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 350


Exercise 7.6 page: 327

Integrate the functions in Exercises 1 to 22.

1. x sin x

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 351

2. x sin 3x

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 352

3. x2 ex

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 353

NCERT Solutions Mathematics Class 12 Chapter 7 - 354

4. x log x

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 355

5. x log 2x

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 356

6. x2 log x

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 357

7. x sin -1 x

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 358

NCERT Solutions Mathematics Class 12 Chapter 7 - 359

8. x tan -1 x

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 360

NCERT Solutions Mathematics Class 12 Chapter 7 - 361

9. x cos -1 x

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 362

NCERT Solutions Mathematics Class 12 Chapter 7 - 363

NCERT Solutions Mathematics Class 12 Chapter 7 - 364

NCERT Solutions Mathematics Class 12 Chapter 7 - 365

10. (sin -1 x)2

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 366

NCERT Solutions Mathematics Class 12 Chapter 7 - 367

11.

NCERT Solutions Mathematics Class 12 Chapter 7 - 368

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 369

NCERT Solutions Mathematics Class 12 Chapter 7 - 370

12. x sec2 x

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 371

13. tan -1 x

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 372

NCERT Solutions Mathematics Class 12 Chapter 7 - 373

14. x (log x)2

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 374

NCERT Solutions Mathematics Class 12 Chapter 7 - 375

15. (x2 + 1) log x

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 376

NCERT Solutions Mathematics Class 12 Chapter 7 - 377

NCERT Solutions Mathematics Class 12 Chapter 7 - 378

16. ex (sin x + cos x)

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 379

17.

NCERT Solutions Mathematics Class 12 Chapter 7 - 380

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 381

NCERT Solutions Mathematics Class 12 Chapter 7 - 382

18.

NCERT Solutions Mathematics Class 12 Chapter 7 - 383

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 384

NCERT Solutions Mathematics Class 12 Chapter 7 - 385

NCERT Solutions Mathematics Class 12 Chapter 7 - 386

19.

NCERT Solutions Mathematics Class 12 Chapter 7 - 387

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 388

20.

NCERT Solutions Mathematics Class 12 Chapter 7 - 389

Solution:

It is given that

NCERT Solutions Mathematics Class 12 Chapter 7 - 390

21. e2x sin x

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 391

NCERT Solutions Mathematics Class 12 Chapter 7 - 392

NCERT Solutions Mathematics Class 12 Chapter 7 - 393

22.

NCERT Solutions Mathematics Class 12 Chapter 7 - 394

Solution:

Take x = tan θ we get dx = sec2 θ dθ

NCERT Solutions Mathematics Class 12 Chapter 7 - 395

NCERT Solutions Mathematics Class 12 Chapter 7 - 396

NCERT Solutions Mathematics Class 12 Chapter 7 - 397

Choose the correct answer in Exercises 23 and 24.

NCERT Solutions Mathematics Class 12 Chapter 7 - 398

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 399

24. ∫ex sec x (1 + tan x) dx equals

(A) ex cos x + C (B) ex sec x + C

(C) ex sin x + C (D) ex tan x + C

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 400

NCERT Solutions Mathematics Class 12 Chapter 7 - 401


EXERCISE 7.7 PAGE NO: 330

Integrate the functions in exercise 1 to 9

NCERT Solutions Mathematics Class 12 Chapter 7 - 402

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 403

NCERT Solutions Mathematics Class 12 Chapter 7 - 404

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 405

NCERT Solutions Mathematics Class 12 Chapter 7 - 406

NCERT Solutions Mathematics Class 12 Chapter 7 - 407

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 408

NCERT Solutions Mathematics Class 12 Chapter 7 - 409

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 410

NCERT Solutions Mathematics Class 12 Chapter 7 - 411

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 412

NCERT Solutions Mathematics Class 12 Chapter 7 - 413

NCERT Solutions Mathematics Class 12 Chapter 7 - 414

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 415

NCERT Solutions Mathematics Class 12 Chapter 7 - 416

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 417

NCERT Solutions Mathematics Class 12 Chapter 7 - 418

NCERT Solutions Mathematics Class 12 Chapter 7 - 419

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 420

NCERT Solutions Mathematics Class 12 Chapter 7 - 421

NCERT Solutions Mathematics Class 12 Chapter 7 - 422

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 423

Choose the correct answer in Exercises 10 to 11

NCERT Solutions Mathematics Class 12 Chapter 7 - 424

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 425

NCERT Solutions Mathematics Class 12 Chapter 7 - 426

NCERT Solutions Mathematics Class 12 Chapter 7 - 427

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 428


EXERCISE 7.8 PAGE NO: 334

Evaluate the following definite integrals as limit of sums.

NCERT Solutions Mathematics Class 12 Chapter 7 - 429

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 430

NCERT Solutions Mathematics Class 12 Chapter 7 - 431

NCERT Solutions Mathematics Class 12 Chapter 7 - 432

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 433
NCERT Solutions Mathematics Class 12 Chapter 7 - 434

NCERT Solutions Mathematics Class 12 Chapter 7 - 435

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 436

NCERT Solutions Mathematics Class 12 Chapter 7 - 437

NCERT Solutions Mathematics Class 12 Chapter 7 - 438

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 439

NCERT Solutions Mathematics Class 12 Chapter 7 - 440

NCERT Solutions Mathematics Class 12 Chapter 7 - 441

= 27/2

NCERT Solutions Mathematics Class 12 Chapter 7 - 442

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 443

NCERT Solutions Mathematics Class 12 Chapter 7 - 444

NCERT Solutions Mathematics Class 12 Chapter 7 - 445

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 446

NCERT Solutions Mathematics Class 12 Chapter 7 - 447

NCERT Solutions Mathematics Class 12 Chapter 7 - 448

NCERT Solutions Mathematics Class 12 Chapter 7 - 449

On computing we get,

f (x) = h (x) + g (x)

= 8 + e8-1


Exercise 7.9 Page No: 338

Evaluate the definite integrals in Exercises 1 to 20.

1.
NCERT Solutions Mathematics Class 12 Chapter 7 - 450

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 451

2. NCERT Solutions Mathematics Class 12 Chapter 7 - 452

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 453

3. NCERT Solutions Mathematics Class 12 Chapter 7 - 454

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 455

NCERT Solutions Mathematics Class 12 Chapter 7 - 466

4. NCERT Solutions Mathematics Class 12 Chapter 7 - 467

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 468

NCERT Solutions Mathematics Class 12 Chapter 7 - 469

5. NCERT Solutions Mathematics Class 12 Chapter 7 - 470

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 471

6. NCERT Solutions Mathematics Class 12 Chapter 7 - 472

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 473

NCERT Solutions Mathematics Class 12 Chapter 7 - 474

7. NCERT Solutions Mathematics Class 12 Chapter 7 - 475

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 476

8. NCERT Solutions Mathematics Class 12 Chapter 7 - 477

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 478

9. NCERT Solutions Mathematics Class 12 Chapter 7 - 479

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 480

NCERT Solutions Mathematics Class 12 Chapter 7 - 481

10. NCERT Solutions Mathematics Class 12 Chapter 7 - 482

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 483

11. NCERT Solutions Mathematics Class 12 Chapter 7 - 484

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 485

NCERT Solutions Mathematics Class 12 Chapter 7 - 486

12. NCERT Solutions Mathematics Class 12 Chapter 7 - 487

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 488

NCERT Solutions Mathematics Class 12 Chapter 7 - 489

13. NCERT Solutions Mathematics Class 12 Chapter 7 - 490

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 491

14. NCERT Solutions Mathematics Class 12 Chapter 7 - 492

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 493

NCERT Solutions Mathematics Class 12 Chapter 7 - 494

NCERT Solutions Mathematics Class 12 Chapter 7 - 495

NCERT Solutions Mathematics Class 12 Chapter 7 - 496

15. NCERT Solutions Mathematics Class 12 Chapter 7 - 497

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 498

NCERT Solutions Mathematics Class 12 Chapter 7 - 499

16. NCERT Solutions Mathematics Class 12 Chapter 7 - 500

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 501

NCERT Solutions Mathematics Class 12 Chapter 7 - 502

NCERT Solutions Mathematics Class 12 Chapter 7 - 503

NCERT Solutions Mathematics Class 12 Chapter 7 - 504

17. NCERT Solutions Mathematics Class 12 Chapter 7 - 505

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 506

NCERT Solutions Mathematics Class 12 Chapter 7 - 507

18. NCERT Solutions Mathematics Class 12 Chapter 7 - 508

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 509

19. NCERT Solutions Mathematics Class 12 Chapter 7 - 510

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 511

NCERT Solutions Mathematics Class 12 Chapter 7 - 512

NCERT Solutions Mathematics Class 12 Chapter 7 - 513

20. NCERT Solutions Mathematics Class 12 Chapter 7 - 514

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 515

NCERT Solutions Mathematics Class 12 Chapter 7 - 516

NCERT Solutions Mathematics Class 12 Chapter 7 - 517

NCERT Solutions Mathematics Class 12 Chapter 7 - 51821.

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 519

Hence, option (D) is correct.

22.

NCERT Solutions Mathematics Class 12 Chapter 7 - 520

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 521

NCERT Solutions Mathematics Class 12 Chapter 7 - 522

Hence, option (C) is correct.


Exercise 7.10 Page No: 338

Evaluate the integrals in Exercise 1 to 8 by substitution.

1. NCERT Solutions Mathematics Class 12 Chapter 7 - 523

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 524

2. NCERT Solutions Mathematics Class 12 Chapter 7 - 525

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 526

NCERT Solutions Mathematics Class 12 Chapter 7 - 527

NCERT Solutions Mathematics Class 12 Chapter 7 - 528

3. NCERT Solutions Mathematics Class 12 Chapter 7 - 529

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 530

NCERT Solutions Mathematics Class 12 Chapter 7 - 531

NCERT Solutions Mathematics Class 12 Chapter 7 - 532

4. NCERT Solutions Mathematics Class 12 Chapter 7 - 533

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 534

NCERT Solutions Mathematics Class 12 Chapter 7 - 535

NCERT Solutions Mathematics Class 12 Chapter 7 - 536

5. NCERT Solutions Mathematics Class 12 Chapter 7 - 537

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 538

NCERT Solutions Mathematics Class 12 Chapter 7 - 539

6. NCERT Solutions Mathematics Class 12 Chapter 7 - 540

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 541

NCERT Solutions Mathematics Class 12 Chapter 7 - 542

NCERT Solutions Mathematics Class 12 Chapter 7 - 543

NCERT Solutions Mathematics Class 12 Chapter 7 - 544

NCERT Solutions Mathematics Class 12 Chapter 7 - 545

7. NCERT Solutions Mathematics Class 12 Chapter 7 - 546

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 547

NCERT Solutions Mathematics Class 12 Chapter 7 - 548

8. NCERT Solutions Mathematics Class 12 Chapter 7 - 549

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 550

NCERT Solutions Mathematics Class 12 Chapter 7 - 551

NCERT Solutions Mathematics Class 12 Chapter 7 - 552

Choose the correct answer in Exercise 9 and 10.

9. NCERT Solutions Mathematics Class 12 Chapter 7 - 553

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 554

NCERT Solutions Mathematics Class 12 Chapter 7 - 555

NCERT Solutions Mathematics Class 12 Chapter 7 - 556

NCERT Solutions Mathematics Class 12 Chapter 7 - 557

NCERT Solutions Mathematics Class 12 Chapter 7 - 558

10. NCERT Solutions Mathematics Class 12 Chapter 7 - 559

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 560


Exercise 7.11 Page No: 347

By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19.

NCERT Solutions Mathematics Class 12 Chapter 7 - 561

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 562

NCERT Solutions Mathematics Class 12 Chapter 7 - 563

NCERT Solutions Mathematics Class 12 Chapter 7 - 564

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 565

NCERT Solutions Mathematics Class 12 Chapter 7 - 566

NCERT Solutions Mathematics Class 12 Chapter 7 - 567

NCERT Solutions Mathematics Class 12 Chapter 7 - 568

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 569

NCERT Solutions Mathematics Class 12 Chapter 7 - 570

NCERT Solutions Mathematics Class 12 Chapter 7 - 571

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 572

NCERT Solutions Mathematics Class 12 Chapter 7 - 573

NCERT Solutions Mathematics Class 12 Chapter 7 - 574

NCERT Solutions Mathematics Class 12 Chapter 7 - 575

NCERT Solutions Mathematics Class 12 Chapter 7 - 576

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 577

NCERT Solutions Mathematics Class 12 Chapter 7 - 578

NCERT Solutions Mathematics Class 12 Chapter 7 - 579

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 580

NCERT Solutions Mathematics Class 12 Chapter 7 - 581

NCERT Solutions Mathematics Class 12 Chapter 7 - 582

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 583

NCERT Solutions Mathematics Class 12 Chapter 7 - 584

NCERT Solutions Mathematics Class 12 Chapter 7 - 585

NCERT Solutions Mathematics Class 12 Chapter 7 - 586

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 587

NCERT Solutions Mathematics Class 12 Chapter 7 - 588

NCERT Solutions Mathematics Class 12 Chapter 7 - 589

NCERT Solutions Mathematics Class 12 Chapter 7 - 590

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 591

NCERT Solutions Mathematics Class 12 Chapter 7 - 592

NCERT Solutions Mathematics Class 12 Chapter 7 - 593

NCERT Solutions Mathematics Class 12 Chapter 7 - 594

NCERT Solutions Mathematics Class 12 Chapter 7 - 595

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 596

NCERT Solutions Mathematics Class 12 Chapter 7 - 597

NCERT Solutions Mathematics Class 12 Chapter 7 - 598

NCERT Solutions Mathematics Class 12 Chapter 7 - 599

NCERT Solutions Mathematics Class 12 Chapter 7 - 600

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 601

NCERT Solutions Mathematics Class 12 Chapter 7 - 602

NCERT Solutions Mathematics Class 12 Chapter 7 - 603

NCERT Solutions Mathematics Class 12 Chapter 7 - 604

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 605

NCERT Solutions Mathematics Class 12 Chapter 7 - 606

NCERT Solutions Mathematics Class 12 Chapter 7 - 607

NCERT Solutions Mathematics Class 12 Chapter 7 - 608

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 609

NCERT Solutions Mathematics Class 12 Chapter 7 - 610

NCERT Solutions Mathematics Class 12 Chapter 7 - 611

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 612

NCERT Solutions Mathematics Class 12 Chapter 7 - 613

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 614

NCERT Solutions Mathematics Class 12 Chapter 7 - 615

NCERT Solutions Mathematics Class 12 Chapter 7 - 616

NCERT Solutions Mathematics Class 12 Chapter 7 - 617

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 618

NCERT Solutions Mathematics Class 12 Chapter 7 - 619

NCERT Solutions Mathematics Class 12 Chapter 7 - 620

NCERT Solutions Mathematics Class 12 Chapter 7 - 621

NCERT Solutions Mathematics Class 12 Chapter 7 - 622

NCERT Solutions Mathematics Class 12 Chapter 7 - 623

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 624

NCERT Solutions Mathematics Class 12 Chapter 7 - 625

NCERT Solutions Mathematics Class 12 Chapter 7 - 626

NCERT Solutions Mathematics Class 12 Chapter 7 - 627

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 628

NCERT Solutions Mathematics Class 12 Chapter 7 - 629

NCERT Solutions Mathematics Class 12 Chapter 7 - 630

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 631

NCERT Solutions Mathematics Class 12 Chapter 7 - 632

Choose the correct answer in Exercises 20 and 21.

NCERT Solutions Mathematics Class 12 Chapter 7 - 633

(A) 0 (B) 2 (C) π (D) 1

Solution:

(C) π

Explanation:

NCERT Solutions Mathematics Class 12 Chapter 7 - 634

NCERT Solutions Mathematics Class 12 Chapter 7 - 635

NCERT Solutions Mathematics Class 12 Chapter 7 - 636

(A) 2 (B) 3/4 (C) 0 (D) –2

Solution:

(C) 0

Explanation:

NCERT Solutions Mathematics Class 12 Chapter 7 - 637

NCERT Solutions Mathematics Class 12 Chapter 7 - 638

NCERT Solutions Mathematics Class 12 Chapter 7 - 639


Miscellaneous Exercise Page No: 352

Integrate the functions in Exercises 1 to 24.

NCERT Solutions Mathematics Class 12 Chapter 7 - 640

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 641

NCERT Solutions Mathematics Class 12 Chapter 7 - 642

NCERT Solutions Mathematics Class 12 Chapter 7 - 643

NCERT Solutions Mathematics Class 12 Chapter 7 - 644

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 645

NCERT Solutions Mathematics Class 12 Chapter 7 - 646

NCERT Solutions Mathematics Class 12 Chapter 7 - 647

NCERT Solutions Mathematics Class 12 Chapter 7 - 648

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 649

NCERT Solutions Mathematics Class 12 Chapter 7 - 650

NCERT Solutions Mathematics Class 12 Chapter 7 - 651

NCERT Solutions Mathematics Class 12 Chapter 7 - 652

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 653

NCERT Solutions Mathematics Class 12 Chapter 7 - 654

NCERT Solutions Mathematics Class 12 Chapter 7 - 655

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 656

NCERT Solutions Mathematics Class 12 Chapter 7 - 657

NCERT Solutions Mathematics Class 12 Chapter 7 - 658

NCERT Solutions Mathematics Class 12 Chapter 7 - 659

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 660

NCERT Solutions Mathematics Class 12 Chapter 7 - 661

NCERT Solutions Mathematics Class 12 Chapter 7 - 662

NCERT Solutions Mathematics Class 12 Chapter 7 - 663

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 664

NCERT Solutions Mathematics Class 12 Chapter 7 - 665

NCERT Solutions Mathematics Class 12 Chapter 7 - 666

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 667

NCERT Solutions Mathematics Class 12 Chapter 7 - 668

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 669

NCERT Solutions Mathematics Class 12 Chapter 7 - 670

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 671

NCERT Solutions Mathematics Class 12 Chapter 7 - 672

NCERT Solutions Mathematics Class 12 Chapter 7 - 673

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 674

NCERT Solutions Mathematics Class 12 Chapter 7 - 675

NCERT Solutions Mathematics Class 12 Chapter 7 - 676

NCERT Solutions Mathematics Class 12 Chapter 7 - 677

NCERT Solutions Mathematics Class 12 Chapter 7 - 678

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 679

NCERT Solutions Mathematics Class 12 Chapter 7 - 680

NCERT Solutions Mathematics Class 12 Chapter 7 - 681

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 682

NCERT Solutions Mathematics Class 12 Chapter 7 - 683

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 684

NCERT Solutions Mathematics Class 12 Chapter 7 - 685

NCERT Solutions Mathematics Class 12 Chapter 7 - 686

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 687

NCERT Solutions Mathematics Class 12 Chapter 7 - 688

NCERT Solutions Mathematics Class 12 Chapter 7 - 689

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 690

NCERT Solutions Mathematics Class 12 Chapter 7 - 691

NCERT Solutions Mathematics Class 12 Chapter 7 - 692

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 693

NCERT Solutions Mathematics Class 12 Chapter 7 - 694

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 695

NCERT Solutions Mathematics Class 12 Chapter 7 - 696

NCERT Solutions Mathematics Class 12 Chapter 7 - 697

NCERT Solutions Mathematics Class 12 Chapter 7 - 698

NCERT Solutions Mathematics Class 12 Chapter 7 - 699

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 700

NCERT Solutions Mathematics Class 12 Chapter 7 - 701

NCERT Solutions Mathematics Class 12 Chapter 7 - 702

NCERT Solutions Mathematics Class 12 Chapter 7 - 703

NCERT Solutions Mathematics Class 12 Chapter 7 - 704

NCERT Solutions Mathematics Class 12 Chapter 7 - 705

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 706

NCERT Solutions Mathematics Class 12 Chapter 7 - 707

NCERT Solutions Mathematics Class 12 Chapter 7 - 708

NCERT Solutions Mathematics Class 12 Chapter 7 - 709

NCERT Solutions Mathematics Class 12 Chapter 7 - 710

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 711

NCERT Solutions Mathematics Class 12 Chapter 7 - 712

NCERT Solutions Mathematics Class 12 Chapter 7 - 713

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 714

NCERT Solutions Mathematics Class 12 Chapter 7 - 715

NCERT Solutions Mathematics Class 12 Chapter 7 - 716

NCERT Solutions Mathematics Class 12 Chapter 7 - 717

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 718

NCERT Solutions Mathematics Class 12 Chapter 7 - 719

NCERT Solutions Mathematics Class 12 Chapter 7 - 720

NCERT Solutions Mathematics Class 12 Chapter 7 - 721

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 722

NCERT Solutions Mathematics Class 12 Chapter 7 - 723

NCERT Solutions Mathematics Class 12 Chapter 7 - 724

NCERT Solutions Mathematics Class 12 Chapter 7 - 725

Evaluate the definite integrals in Exercises 25 to 33.

NCERT Solutions Mathematics Class 12 Chapter 7 - 726

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 727

NCERT Solutions Mathematics Class 12 Chapter 7 - 728

NCERT Solutions Mathematics Class 12 Chapter 7 - 729

NCERT Solutions Mathematics Class 12 Chapter 7 - 730

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 731

NCERT Solutions Mathematics Class 12 Chapter 7 - 732

NCERT Solutions Mathematics Class 12 Chapter 7 - 733

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 734

NCERT Solutions Mathematics Class 12 Chapter 7 - 735

NCERT Solutions Mathematics Class 12 Chapter 7 - 736

NCERT Solutions Mathematics Class 12 Chapter 7 - 737

NCERT Solutions Mathematics Class 12 Chapter 7 - 738

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 739

NCERT Solutions Mathematics Class 12 Chapter 7 - 740

NCERT Solutions Mathematics Class 12 Chapter 7 - 741

NCERT Solutions Mathematics Class 12 Chapter 7 - 742

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 743

NCERT Solutions Mathematics Class 12 Chapter 7 - 744

NCERT Solutions Mathematics Class 12 Chapter 7 - 745

NCERT Solutions Mathematics Class 12 Chapter 7 - 746

NCERT Solutions Mathematics Class 12 Chapter 7 - 747

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 748

NCERT Solutions Mathematics Class 12 Chapter 7 - 749

NCERT Solutions Mathematics Class 12 Chapter 7 - 750

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 751

NCERT Solutions Mathematics Class 12 Chapter 7 - 752

NCERT Solutions Mathematics Class 12 Chapter 7 - 753

NCERT Solutions Mathematics Class 12 Chapter 7 - 754

NCERT Solutions Mathematics Class 12 Chapter 7 - 755

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 756

NCERT Solutions Mathematics Class 12 Chapter 7 - 757

NCERT Solutions Mathematics Class 12 Chapter 7 - 758

NCERT Solutions Mathematics Class 12 Chapter 7 - 759

NCERT Solutions Mathematics Class 12 Chapter 7 - 760

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 761

NCERT Solutions Mathematics Class 12 Chapter 7 - 762

NCERT Solutions Mathematics Class 12 Chapter 7 - 763

NCERT Solutions Mathematics Class 12 Chapter 7 - 764

NCERT Solutions Mathematics Class 12 Chapter 7 - 765

NCERT Solutions Mathematics Class 12 Chapter 7 - 766

Prove the following (Exercises 34 to 39)

NCERT Solutions Mathematics Class 12 Chapter 7 - 767

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 768

NCERT Solutions Mathematics Class 12 Chapter 7 - 769

NCERT Solutions Mathematics Class 12 Chapter 7 - 770

NCERT Solutions Mathematics Class 12 Chapter 7 - 771

NCERT Solutions Mathematics Class 12 Chapter 7 - 772

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 773

NCERT Solutions Mathematics Class 12 Chapter 7 - 774

NCERT Solutions Mathematics Class 12 Chapter 7 - 775

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 776

NCERT Solutions Mathematics Class 12 Chapter 7 - 777

NCERT Solutions Mathematics Class 12 Chapter 7 - 778

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 779

NCERT Solutions Mathematics Class 12 Chapter 7 - 780

NCERT Solutions Mathematics Class 12 Chapter 7 - 781

NCERT Solutions Mathematics Class 12 Chapter 7 - 782

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 783

NCERT Solutions Mathematics Class 12 Chapter 7 - 784

NCERT Solutions Mathematics Class 12 Chapter 7 - 785

Hence the proof.

NCERT Solutions Mathematics Class 12 Chapter 7 - 786

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 787

NCERT Solutions Mathematics Class 12 Chapter 7 - 788

NCERT Solutions Mathematics Class 12 Chapter 7 - 789

NCERT Solutions Mathematics Class 12 Chapter 7 - 790

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 791

NCERT Solutions Mathematics Class 12 Chapter 7 - 792

NCERT Solutions Mathematics Class 12 Chapter 7 - 793

Choose the correct answers in Exercises 41 to 44.

NCERT Solutions Mathematics Class 12 Chapter 7 - 794

NCERT Solutions Mathematics Class 12 Chapter 7 - 795

Solution:

(A) tan-1 (ex) + C

Explanation:

NCERT Solutions Mathematics Class 12 Chapter 7 - 796

NCERT Solutions Mathematics Class 12 Chapter 7 - 797

Solution:

(B) log |sin x + cos x| + C

Explanation:

NCERT Solutions Mathematics Class 12 Chapter 7 - 798

NCERT Solutions Mathematics Class 12 Chapter 7 - 799

Solution:

NCERT Solutions Mathematics Class 12 Chapter 7 - 800

Explanation:

NCERT Solutions Mathematics Class 12 Chapter 7 - 801

Hence, correct option is (D).

NCERT Solutions Mathematics Class 12 Chapter 7 - 802

(A) 1 (B) 0 (C) –1 (D) π

Solution:

(B) 0

Explanation:

NCERT Solutions Mathematics Class 12 Chapter 7 - 803

NCERT Solutions Mathematics Class 12 Chapter 7 - 804


The major concepts of Maths covered in the Chapter 7- Integrals of NCERT Solutions for Class 12 includes:

7.1 Introduction

7.2 Integration as an Inverse Process of Differentiation

7.2.1 Geometrical interpretation of indefinite integral

7.2.2 Some properties of indefinite integral

7.2.3 Comparison between differentiation and integration

7.3 Methods of Integration

7.3.1 Integration by substitution

7.3.2 Integration using trigonometric identities

7.4 Integrals of Some Particular Functions

7.5 Integration by Partial Fractions

7.6 Integration by Parts

7.7 Definite Integral

7.7.1 Definite integral as the limit of a sum

7.8 Fundamental Theorem of Calculus

7.8.1 Area function

7.8.2 First fundamental theorem of integral calculus

7.8.3 Second fundamental theorem of integral calculus

7.9 Evaluation of Definite Integrals by Substitution

7.10 Some Properties of Definite Integrals

Exercise 7.1 Solutions 22 Questions

Exercise 7.2 Solutions 39 Questions

Exercise 7.3 Solutions 24 Questions

Exercise 7.4 Solutions 25 Questions

Exercise 7.5 Solutions 23 Questions

Exercise 7.6 Solutions 24 Questions

Exercise 7.7 Solutions 11 Questions

Exercise 7.8 Solutions 6 Questions

Exercise 7.9 Solutions 22 Questions

Exercise 7.10 Solutions 10 Questions

Exercise 7.11 Solutions 21 Questions

Miscellaneous Exercise On Chapter 7 Solutions 44 Questions

NCERT Solutions for Class 12 Maths Chapter 7- Integrals

The chapter integrals belongs to the unit Calculus, that adds up to 35 marks of the total marks. There are 11 exercises along with a miscellaneous exercise in this chapter to help the students get thorough with the concept of Integrals. The Chapter 7 of NCERT Solutions for Class 12 Maths discusses the following:

  1. Integration is the inverse process of differentiation. In the differential calculus, we are given a function and we have to find the derivative or differential of this function, but in the integral calculus, we are to find a function whose differential is given. Thus, integration is a process which is the inverse of differentiation.
  2. From the geometric point of view, an indefinite integral is collection of family of curves, each of which is obtained by translating one of the curves parallel to itself upwards or downwards along the y-axis.
  3. Some properties of indefinite integrals.
  4. Some standard integrals
  5. Integration by partial fractions
  6. Integration by substitution: A change in the variable of integration often reduces an integral to one of the fundamental integrals. The method in which we change the variable to some other variable is called the method of substitution. When the integrand involves some trigonometric functions, we use some well known identities to find the integrals.
  7. Integration by parts: the integral of the product of two functions = first function × integral of the second function – integral of {differential coefficient of the first function × integral of the second function}.
  8. First fundamental theorem of integral calculus.
  9. Second fundamental theorem of integral calculus.

Key Features of NCERT Solutions for Class 12 Maths Chapter 7- Integrals

Learning the chapter Integrals enables the students to study the following:

Integration as the inverse process of differentiation.Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.

Key Features of NCERT Solutions for Class 12 Maths Chapter 7- Integrals

Definite integrals as a limit of a sum, Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

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