# NCERT Solutions for class 12th Mathematics Chapter 7 Integrals

#### Exercise 7.1

Question. Find an antiderivative (or integral) of the following by the method of inspection :

Note: The process of differentiation and integration are inverse of each other

Find the following integrals in Exercises 6 to 20 :

Note: Antiderivative or integral of a function is not unique. In actual, there exist infinitely many antiderivatives of any function that can be obtained by choosing ‘C’ arbitrarily from the set of real numbers and ‘C’ is called the constant of integration. For this reason ‘C’ is written in the final answer.

Question. Choose the correct answer in Questions 21 and 22.

#### Exercise 7.2

Note: In this exercise, we will be using the substitution method to find the integrals of functions given by converting them into standard forms.

Integrate the functions in questions 1 to 37 :

Note: If in case of f (x) × g(x), f (x) is the differentiation of g(x), then in order to solve it we substitute g(x) by some variable say (t)

Question. Choose the correct answer in questions 38 and 39:

#### Exercise 7.3

Note: In this exerices we will use trigonometric identities to find the integrals of function given by converting them into standard forms.

Find the integrals of the functions in questions 1 to 22

Question. Choose the correct answer in questions 23 and 24

#### Exercise 7.4

Question. Integrate the functions in questions 1 to 23

Question. Choose the correct answer in questions 24 and 25 :

#### Exercise 7.5

Note: In this exercise we will use partial fractions to find the integrals of given functions by converting them into the standard forms

Question. Integrate the rational functions in questions 1 to 21

Question. Choose the correct answer in each of the following :

#### Exercise 7.6

Question. Integrate the functions in questions 1 to 22 .

#### Exercise 7.7

Question. Integral the functions in questions 1 to 9

Question. Choose the correct answer in the questions 10 to 11:

#### Exercise 7.8

Question. Evaluate the following definite integral as limit of sums

#### Exercise 7.9

Question. Evaluate the definite integrals in questions 1 to 20

Question 21. Choose the correct answer in questions 21 and 22 :

#### Exercise 7.10

Question. Evaluate the integrals in questions 1 to 8 using substitution.

Question. Choose the correct answer in questions 9 and 10

#### Exercise 7.11

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

#### Miscellaneous Exercise

Question. Integrate the functions from questions 1 to 24.

Evaluate the following definite integral from questions 25 to 33

Prove the following from questions 34 to 39

Question 40. Evaluate:

Question. Choose the correct answers in questions 41 to 44

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