Important Questions for Class 12 Maths Chapter 7 - Integrals

Important questions for class 12 Maths Chapter 7 – Integrals are provided here. While preparing for class 12 Maths board examination, students should be thorough with all the concepts. If they are clear in the concepts, they should be capable of solving any question. For that, students need rigorous practice. Here, the important questions are taken as per the CBSE Board syllabus. Practice these important questions for class 12 maths chapter 7 integrals multiple times, to avoid the mistakes in the final examination. Get all the important questions for Class 12th Maths chapters provided at BYJU’S.

Class 12 chapter 7 – Integrals covers important concepts such as integrations, definite and indefinite integrals, some properties of definite integrals, fundamental theorem of calculus, and also, the methods of integration such as:

• Integration by parts
• Integration by substitution
• Integration using partial fractions

Also, check:

• Important 4 Marks Questions for CBSE Class 12 Maths
• Important 6 marks Questions for CBSE Class 12 Maths

Class 12 Chapter 7 Integrals Important Questions with Solutions

The most important questions from class 12 Maths Chapter 7 – Integrals are provided below.

Question 1:

Evaluate: ∫ 3ax/(b² +c²x²) dx

Solution:

To evaluate the integral, I = ∫ 3ax/(b² +c²x²) dx

Let us take v = b² +c²x², then

dv = 2c²x dx

Thus, ∫ 3ax/(b² +c²x²) dx

= (3ax/2c²x)∫dv/v

Now, cancel x on both numerator and denominator, we get

= (3a/2c²)∫dv/v

= (3a/2c²) log |b² +c²x²| + C

Where C is an arbitrary constant

Question 2:

Determine ∫tan⁸x sec⁴ x dx

Solution:

Given: ∫tan⁸x sec⁴ x dx

Let I = ∫tan⁸x sec⁴ x dx — (1)

Now, split sec⁴x = (sec²x) (sec²x)

Now, substitute in (1)

I = ∫tan⁸x (sec²x) (sec²x) dx

= ∫tan⁸x (tan² x +1) (sec²x) dx

It can be written as:

= ∫tan¹⁰x sec² x dx + ∫tan⁸x sec² x dx

Now, integrate the terms with respect to x, we get:

I =( tan¹¹ x /11) + ( tan⁹ x /9) + C

Hence, ∫tan⁸x sec⁴ x dx = ( tan¹¹ x /11) + ( tan⁹ x /9) + C

Question 3:

Write the anti-derivative of the following function: 3x²+4x³

Solution:

Given: 3x²+4x³

The antiderivative of the given function is written as:

∫3x²+4x³ dx =  3(x³/3) + 4(x⁴/4)

= x³ + x⁴

Thus, the antiderivative of 3x²+4x³ = x³ + x⁴

Question 4:

Determine the antiderivative F of “f” , which is defined by f (x) = 4x³ – 6, where F (0) = 3

Solution:

Given function: f (x) = 4x³ – 6

Now, integrate the function:

∫4x³ – 6 dx = 4(x⁴/4)-6x + C

∫4x³ – 6 dx = x⁴ – 6x + C

Thus, the antiderivative of the function, F is x⁴ – 6x + C, where C is a constant

Also, given that, F(0) = 3,

Now, substitute x = 0 in the obtained antiderivative function, we get:

(0)⁴ – 6(0) + C = 3

Therefore, C = 3.

Now, substitute C = 3 in antiderivative function

Hence, the required antiderivative function is x⁴ – 6x + 3.

Question 5:

Integrate the given function using integration by substitution: 2x sin(x²+ 1) with respect to x:

Solution:

Given function: 2x sin(x²+ 1)

We know that, the derivative of x² + 1 is 2x.

Now, use the substitution method, we get

x² + 1 = t, so that 2x dx = dt.

Hence, we get ∫ 2x sin ( x² +1) dx = ∫ sint dt

= – cos t + C

= – cos (x² + 1) + C

Where C is an arbitrary constant

Therefore, the antiderivative of 2x sin(x²+ 1) using integration by substitution method is = – cos (x² + 1) + C

Question 6:

Integrate: ∫ sin³ x cos²x dx

Solution:

Given that, ∫ sin³ x cos²x dx

This can be written as:

∫ sin³ x cos²x dx = ∫ sin² x cos²x (sin x) dx

=∫(1 – cos²x ) cos²x (sin x) dx —(1)

Now, substitute t = cos x,

Then dt = -sin x dx

Now, equation can be written as:

Thus, ∫ sin³ x cos²x dx = – ∫ (1-t²)t² dt

Now, multiply t² inside the bracket, we get

= – ∫ (t²-t⁴) dt

Now, integrate the above function:

= – [(t³/3) – (t⁵/5)] + C —(2)

Where C is a constant

Now, substitute t = cos x in (2)

= -(⅓)cos³x +(1/5)cos⁵x + C

Hence, ∫ sin³ x cos²x dx = -(⅓)cos³x +(1/5)cos⁵x + C

Practice Problems for Class 12 Maths Chapter 7

Solve the practice problem given below:

1. Integrate the function using integration by substution method: ∫1/(1 + tan x) dx
2. Find ∫sin 2x cos 3x dx.
3. Evaluate the integral (cos 2x+ 2 sin²x)/cos²x
4. Evaluate the integral: ∫ dx/(x²-16)
5. Find ∫ dx/((x+1)(x+2)) using integration by partial fractions.

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