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CBSE - Class 12 Mathematics Integrals Worksheet

Prove the following (Exercises 32 to 37)
36. $\int_{0}^{\frac{\pi}{4}} 2\tan^3 x dx = 1 - \log 2$

Integrate the functions in Exercises 1 to 23.
13. $\frac{e^x}{(1+e^x)(2+e^x)}$

Evaluate the definite integrals in Exercises 24 to 31.
29. $\int_{0}^{\frac{\pi}{4}} \frac{\sin x + \cos x}{9+16\sin 2x} dx$

Integrate the functions in Exercises 1 to 23.
1. $\frac{1}{x-x^3}$

Integrate the rational functions in Exercises 1 to 21.
7. $\frac{x}{(x^2+1)(x-1)}$

Choose the correct answer in Exercises 20 and 21.
20. The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} (x^3 + x \cos x + \tan^5 x + 1) dx$ is

a. 0 b. 2 c. $\pi$ d. 1

Choose the correct answer in Exercises 23 and 24.
24. $\int e^x \sec x (1 + \tan x) dx$ equals

a. $e^x \cos x + C$ b. $e^x \sec x + C$ c. $e^x \sin x + C$ d. $e^x \tan x + C$

Choose the correct answer in Exercises 38 and 39.
39. $\int \frac{dx}{\sin^2 x \cos^2 x}$ equals

a. $\tan x + \cot x + C$ b. $\tan x - \cot x + C$ c. $\tan x \cot x + C$ d. $\tan x - \cot 2x + C$

Choose the correct answer in each of the Exercises 22 and 23.
23. $\int \frac{dx}{x(x^2+1)}$ equals

a. $\log |x| - \frac{1}{2} \log (x^2+1) + C$ b. $\log |x| + \frac{1}{2} \log (x^2+1) + C$ c. $- \log |x| + \frac{1}{2} \log (x^2+1) + C$ d. $\frac{1}{2} \log |x| + \log (x^2+1) + C$

10.

Evaluate the definite integrals in Exercises 1 to 20.
5. $\int_{0}^{\frac{\pi}{2}} \cos 2x dx$

11.

Integrate the functions in Exercises 1 to 37:
15. $\frac{x}{9-4x^2}$

12.

Integrate the rational functions in Exercises 1 to 21.
14. $\frac{3x-1}{(x+2)^2}$

13.

Integrate the functions in Exercises 1 to 9.
7. $\sqrt{1+3x-x^2}$

14.

Find the following integrals 14. $\int (1 - x)\sqrt{x} dx$

15.

Integrate the rational functions in Exercises 1 to 21.
2. $\frac{1}{x^2-9}$

16.

Find the integrals of the functions in Exercises 1 to 22:
11. $\cos^4 2x$

17.

By using the properties of definite integrals, evaluate the integrals in Exercises 1 to 19.
14. $\int_{0}^{2\pi} \cos^5 x dx$

18.

Integrate the functions in Exercises 1 to 37:
33. $\frac{1}{1-\tan x}$

19.

Integrate the functions in Exercises 1 to 23.
10. $\frac{1}{\sqrt{x^2+2x+2}}$

20.

Integrate the functions in Exercises 1 to 22.
7. $x \sin^{-1} x$

Worksheet Answers

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