NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.2

Class 12 Maths Chapter 7 Exercise 7.2:- Chapter 7 in Class 12 Maths covers "Integrals." Exercise 7.2 focuses on the evaluation of integrals using various methods, such as substitution, integration by parts, and partial fractions. This exercise helps students understand how to find the integral of different types of functions and improve their problem-solving skills in calculus. The problems in this exercise are designed to provide a comprehensive understanding of the integration techniques and their applications. Check out the NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.2  below.

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NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.2 

Get the NCERT Solutions for Class 12 Maths Chapter 7 Exercise 7.2 below:-

Integrate the functions in Exercise 1 to 8.

Question 1.

Solution :

Let 1 + x2 = t

∴2x dx = dt

Question 2.

Solution :

Let log |x| = t

∴ 1/x dx = dt

Check out: NCERT Solutions for Class 12 Maths Chapter 6 Exercise 6.1

Question 3.

Solution :

Check out: NCERT Solutions for Class 12 Maths Chapter 6 Exercise 6.2

Question 4. sin x ⋅ sin (cos x)

Solution :

sin x ⋅ sin (cos x)

Let cos x = t

∴ −sin x dx = dt

Question 5. sin(ax + b) cos(ax + b)

Solution :

Questionc 6. √ax + b

Solution :

Let ax + b = t

⇒ adx = dt

Question 7. x√x + 2

Solution :

Let  (x + 2) = t

∴ dx = dt

Question 8. x√1 + 2x2
Solution :

Let 1 + 2x2 = t

∴ 4xdx = dt

Integrate the functions in Exercise 9 to 17.

Question 9.

Solution :

Question 10.

Solution :

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Question 11.

Solution :

Question 12.

Solution :

Let x3 – 1 = t

∴ 3x2 dx = dt

Question 13.

Solution :

Let 2 + 3x3 = t

∴ 9x2 dx = dt

Question 14.
Solution :

Let log x = t

∴ 1/x dx = dt

Question 15. x/9 – 4x2

Solution :

Let 9 – 4x2 =  t

∴ −8x dx = dt

Question 16.

Solution :

Let 2x + 3 = t

∴ 2dx = dt

Question 17.
Solution :

Let x2 = t

∴ 2xdx = dt

Integrate the functions in Exercise 18 to 26.

Question 18.

Solution :

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Question 19.

Solution :

Dividing numerator and denominator by ex, we obtain

Question 20.
Solution :

Question 21. tan2 (2x – 3)

Solution :

Question 22. sec2 (7 – 4x)

Solution :

Let 7 − 4x = t

∴ −4dx = dt

Question 23.

Solution :

Question 24.

Solution :

Question 25.

Solution :

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Question 26.

Solution :

Let √x = t

Integrate the functions in Exercise 27 to 37.

Question 27.

Solution :

Let sin 2x = t

Question 28.

Solution :

Let 1 + sin x = t

∴ cos x dx = dt

Question 29. cot x log sin x

Solution :

Let log sin x = t

Question 30. sin x/1 + cos x

Solution :

Let 1 + cos x = t

∴ −sin x dx = dt

Question 31.  sin x/(1 + cos x)2

Solution :

Let 1 + cos x = t

∴ −sin x dx = dt

Question 32. 1/1 + cot x
Solution :

Question 33. 1/1 – tan x

Solution :

Question 34.

Solution :

Question 35.

Solution :

Let 1 + log x = t

∴ 1/x dx = dt

Question 36.

Solution :

Question 37.

Solution :

Let x4 = t

∴ 4x3 dx = dt

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Choose the correct answer in Exercise 38 and 39.

Question 38. equals

(A) 10x – x10 + C
(B) 10x + x10 + C
(C) (10x – x10)-1 + C
(D) log(10x + x10) + C

Solution :

Therefore, option (D) is correct.
Question 39.equals

(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 :

Therefore, option (B) is correct.

Class 12 Maths Chapter 7 Exercise 7.2 Summary

Chapter 7 of Class 12 Maths deals with "Integrals," and Exercise 7.2 specifically focuses on integrating functions using different techniques. Here is a summary of Exercise 7.2:

1. Integration by Substitution: This technique involves substituting a part of the integral with a new variable to simplify the integral into a more manageable form. The goal is to make the integral easier to evaluate.

2. Integration by Parts: This method is used when the integral is a product of two functions. It is based on the formula:
   ∫u dv=uv−∫v du\int u \, dv = uv - \int v \, du∫udv=uv−∫vdu
   where uuu and dvdvdv are parts of the original integral.

3. Partial Fractions: This approach is used to integrate rational functions by expressing them as a sum of simpler fractions, making it easier to integrate each term individually.

4. Special Integrals: Exercise 7.2 also includes problems that involve integrals of specific forms that have standard solutions, such as ∫1a2+x2 dx\int \frac{1}{a^2 + x^2} \, dx∫a2+x21​dx and ∫1a2−x2 dx\int \frac{1}{\sqrt{a^2 - x^2}} \, dx∫a2−x2​1​dx.

Class 12 Maths Chapter 7 Exercise 7.2 FAQs

Q1. What is the primary focus of Exercise 7.2 in Chapter 7?

Ans. Exercise 7.2 primarily focuses on the evaluation of integrals using techniques like substitution, integration by parts, and partial fractions.

Q2. How do you determine when to use substitution in an integral?

Ans. Substitution is useful when the integral contains a function and its derivative. By substituting the inner function with a new variable, the integral can be simplified.

Q3. What is the formula for integration by parts?

Ans. The formula is ∫u dv=uv−∫v du\int u \, dv = uv - \int v \, du∫udv=uv−∫vdu, where uuu and dvdvdv are parts of the original integral.

Q4. When should you use integration by parts?

Ans. Integration by parts is typically used when integrating the product of two functions, especially if one function is easily differentiable and the other is easily integrable.

Q5. What are partial fractions and when are they used in integration?

Ans. Partial fractions involve breaking down a complex rational function into simpler fractions. This method is used when integrating rational functions to simplify the integration process.