## RD Sharma Solutions for Class 12 Maths Chapter 19 – Free PDF Download

**RD Sharma Solutions for Class 12 Maths Chapter 19 –Â Indefinite Integrals**Â is given here. By solving exercise-wiseÂ problems usingÂ RD Sharma Solutions for Class 12Â daily helps students improve their problem solving and logical thinking skills, which are important to achieve a better academic score. The main aim is to help students self analyze the areas, which require more practice from the exam perspective. With the help of **RD Sharma Solutions**, students can now solve the exercise problems in a shorter duration with a clear idea about the concepts.

The 19^{th} Chapter, Indefinite Integrals of RD Sharma Solutions for Class 12 Maths explains some standard results on integration along with fundamental integration formulae.Â TheÂ **RD Sharma Solutions for Class 12** are formulated by BYJUâ€™S experts to provide a fundamental aspect of Maths, which in turn, assists students to understand every concept clearly.Â The solutions PDF is a major reference guide to help students score well in the Class 12 examination.

## Download the PDF of RD Sharma Solutions For Class 12 Maths Chapter 19 Indefinite Integrals

### Access answers to Maths RD Sharma Solutions For Class 12 Chapter 19 – Indefinite Integrals

### Exercise 19.1 Page No: 19.4

**1. Evaluate the following integrals:**

**Solution:**

Given

**Solution:**

Given

**Solution:**

Given

**Solution:**

Given

**Solution:**

Given

**Solution:**

Given

Solution:

Given

**Solution:**

Given

**2. Evaluate:**

**Solution:**

Given

**Solution:**

Given

**3. Evaluate:**

**Solution:**

Given

### Exercise 19.2 Page No: 19.14

**Evaluate the following integrals (1 – 44):**

**Solution: **

**Solution:**

Given

**Solution:**

Given

**Solution:**

Given,

âˆ«(2 â€“ 3x)(3 + 2x)(1 â€“ 2x) dx

= âˆ«(6 + 4x â€“ 9x â€“ 6x^{2})(1 â€“ 2x) dx

= âˆ«(6 â€“ 5x â€“ 6x^{2})(1 â€“ 2x) dx

= âˆ«(6 â€“ 5x â€“ 6x^{2} â€“ 12x + 10x^{2} + 12x^{3}) dx

= âˆ«(6 â€“ 17x + 4x^{2} + 12x^{3}) dx

Upon splitting the above, we have

= âˆ«6 dx â€“ âˆ«17x dx + âˆ«4x^{2} dx + âˆ«12x^{3} dx

On integrating using formula,

âˆ«x^{n} dx = x^{n+1}/n+1

we get

= 6x â€“ 17/(1+1) x^{1+1} + 4/(2+1) x^{2+1} + 12/(3+1) x^{3+1} + c

= 6x â€“ 17x^{2}/2 + 4x^{3}/3 + 3x^{4} + c

**Solution:**

Given

**Solution:**

**Solution:**

Given

**Solution:**

Given

**Solution:**

**Solution:**

Given

**Solution:**

Given

**Solution:**

**Solution:**

Given

**Solution:**

Given

**Solution:**

Given

**Solution:**

Given

### Exercise 19.3 Page No: 19.23

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

### Exercise 19.4 Page No: 19.30

**Solution:**

**Solution:**

**Solution:**

### Exercise 19.5 Page No: 19.33

**Solution:**

Given

**Solution:**

**Solution:**

**Solution:**

**Solution:**

### Exercise 19.6 Page No: 19.36

**Solution:**

**Solution:**

**Solution:**

**Solution:**

### Exercise 19.7 Page No: 19.38

**Integrate the following integrals:**

**Solution:**

**Solution:**

**Solution:**

### Exercise 19.8 Page No: 19.47

**Evaluate the following integrals:**

**Solution:**

**Solution:**

Given,

**Solution:**

Given,

**Solution:**

**Solution:**

**Solution:**

Therefore,

= cos (b – a)x + sin(b – a) log |sin(x – b)| + c, where c is an arbitrary constant.

### Exercise 19.9 Page No: 19.57

**Evaluate the following integrals:**

** dx**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

### Exercise 19.10 Page No: 19.65

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

### Exercise 19.11 Page No: 19.69

**Evaluate the following integrals:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

### Exercise 19.12 Page No: 19.73

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

### Exercise 19.13 Page No: 19.79

**Solution:**

**Solution:**

### Exercise 19.14 Page No: 19.83

**Evaluate the following integrals:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

### Exercise 19.15 Page No: 19.86

**Solution:**

**Solution:**

**Solution:**

By using,

**Solution:**

**Solution:**

### Exercise 19.16 Page No: 19.90

**Evaluate the following integrals:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

### Exercise 19.17 Page No: 19.93

**Evaluate the following integrals:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

### Exercise 19.18 Page No: 19.98

**Evaluate the following integrals:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

Let sin x = t

**Solution:**

**Solution:**

**Solution:**

**Solution:**

### Exercise 19.19 Page No: 19.104

**Evaluate the following integrals:**

**Solution:**

We will solve I_{1}Â and I_{2}Â individually.

**Solution:**

**Solution:**

**Solution:**

**Solution:**

### Exercise 19.20 Page No: 19.106

**Evaluate the following integrals:**

**Solution:**

**Solution:**

â‡’Â 1 = (A + B) x + (3A â€“ 2B)

â‡’Â Then A + B = 0 â€¦ (1)

And 3A â€“ 2B = 1 â€¦ (2)

Solving (1) and (2),

2 Ã— (1)Â â†’Â 2A + 2B = 0

1 Ã— (2)Â â†’Â 3A â€“ 2B = 1

5A = 1

âˆ´Â A = 1/5

Substituting A value in (1),

Or I = log|(x – 2)/(x + 3)| + x + c

**Solution:**

**Solution:**

**Solution:**

Hence,

### Exercise 19.21 Page No: 19.110

**Evaluate the following integrals:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

### Exercise 19.22 Page No: 19.114

**Evaluate the following integrals:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

### Exercise 19.23 Page No: 19.117

**Evaluate the following integrals:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

5.

**Solution:**

### Exercise 19.24 Page No: 19.122

**Evaluate the following integrals:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

### Exercise 19.25 Page No: 19.133

**Evaluate the following integrals:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

### Exercise 19.26 Page No: 19.143

**Evaluate the following integrals:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

### Exercise 19.27 Page No: 19.149

**Evaluate the following integrals:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

### Exercise 19.28 Page No: 19.154

**Evaluate the following integrals:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

### Exercise 19.29 Page No: 19.158

**Evaluate the following integrals:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

### Exercise 19.30 Page No: 19.176

**Evaluate the following integrals:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

### Exercise 19.31 Page No: 19.190

**Evaluate the following integrals:**

**Solution:**

The given equation can be written as,

**Solution:**

Now, substituting t as x â€“ 1/x and z as x + 1/x we have

**Solution:**

**Solution:**

We get,

**Solution:**

### Exercise 19.32 Page No: 19.196

**Evaluate the following integrals:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

### Access all the exercises of RD Sharma Solutions For Class 12 Chapter 19 â€“ Indefinite Integrals

## RD Sharma Class 12 Maths Solutions Chapter 19Â Indefinite Integrals

Some of the essential topics covered in this chapter are listed below.

- Definition of primitive or antiderivative
- Definition and meaning of indefinite integral
- Fundamental integration formulae
- Some standard results on integration along with the corollary
- Integration of trigonometric functions
- Integration of exponential functions
- Miscellaneous problems
- Geometrical interpretation of indefinite integral
- Comparison between differentiation and integration
- Methods of integration
- Integration by substitution
- Some standard results
- Evaluation of integrals by using trigonometric substitutions
- Some special integrals
- Integration by parts
- Some important integrals along with theorems
- Integration of rational algebraic functions by using partial fractions
- When the denominator is expressible as a product of distinct linear factors
- When the denominator contains some repeating linear factors
- The denominator contains irreducible quadratic factors

- Integration of some special irrational algebraic functions