Integrals: Class 12 Important Questions with Answers

Practice set: test yourself

Try every question before looking at the answers below. 64 or more out of 80 means you are ready for the board.

C. Very short answer (1 mark each)

1. $\int \cos (3x+1)dx$.
2. $\int \frac{dx}{2x+3}$.
3. ${\int }_0^1{e}^{2x}dx$.
4. $\int {\sin }^{2}xdx$.
5. $\int \frac{x}{1+x^2}dx$.
6. ${\int }_0^{\pi /4}{\sec }^{2}xdx$.
7. $\int \frac{1}{x\log x}dx$.
8. ${\int }_{-\pi }^{\pi }{\sin }^{3}xdx$.
9. $\int \frac{dx}{\sqrt{4-x^2}}$.
10. ${\int }_0^2|x-1|dx$.

D. Short answer I (2 marks each)

1. Find $\int \frac{\sin x}{1+{\cos }^{2}x}dx$.
2. Find $\int x\cos xdx$.
3. Evaluate ${\int }_0^1\frac{x}{\sqrt{1+x^2}}dx$.
4. Find $\int \frac{dx}{x^2-4x+8}$.
5. Find $\int \frac{e^x(1+x)}{(2+x{)}^2}dx$.
6. Evaluate ${\int }_0^{\pi /2}\frac{{\cos }^{3}x}{{\sin }^{3}x+{\cos }^{3}x}dx$.
7. Find $\int \frac{dx}{(x+1)(x+2)}$.
8. Evaluate ${\int }_{-2}^2(x^5+x^3+1)dx$.

Rough work

E. Short answer II (3 marks each)

1. Find $\int \frac{2x+1}{x^2+2x+5}dx$.
2. Find $\int x^2\log xdx$.
3. Evaluate ${\int }_0^{\pi }\frac{x\sin x}{1+{\cos }^{2}x}dx$.
4. Find $\int \frac{x}{(x-1)(x-2)(x-3)}dx$.
5. Find $\int e^x\sin 2xdx$.
6. Evaluate ${\int }_0^{\pi /2}\log \tan xdx$.

F. Long answer (5 marks each)

1. (a) Find $\int \frac{x^2+1}{(x-1{)}^2(x+3)}dx$. (b) Evaluate ${\int }_0^{\pi /4}\frac{dx}{\cos x+\sin x}$.
2. (a) Evaluate ${\int }_0^{\pi }\frac{x}{a^2{\cos }^{2}x+b^2{\sin }^{2}x}dx$. (b) Find $\int \sqrt{x^2+2x+5}dx$.

G. HOTS (higher-order thinking)

1. Why is ${\int }_{-1}^1\frac{dx}{x^2}$ not equal to ${\left[-\frac{1}{x}\right]}_{-1}^1=-2$?
2. Show ${\int }_0^{2a}f(x)dx=0$ if $f(2a-x)=-f(x)$, with an example.
3. Which is larger: ${\int }_0^{1}xdx$ or ${\int }_0^1{x}^{2}dx$? Why?
4. Find $\int \frac{dx}{\sin x+\sqrt{3}\cos x}$.

20 MCQs

1. $\int \frac{dx}{x}$
   (a) $x$ (b) $-\frac{1}{x^2}$ (c) $\log |x|$ (d) $e^x$
2. $\int \sec x\tan xdx$
   (a) $\sec x$ (b) $\tan x$ (c) ${\sec }^{2}x$ (d) $\log |\sec x|$
3. ${\int }_0^{\pi /2}\cos xdx$
   (a) 0 (b) $\frac{\pi }{2}$ (c) $-1$ (d) 1
4. $\int \frac{dx}{x^2+9}$
   (a) ${\tan }^{-1}\frac{x}{3}$ (b) $\frac{1}{3}{\tan }^{-1}\frac{x}{3}$ (c) $3{\tan }^{-1}3x$ (d) $\frac{1}{9}{\tan }^{-1}x$
5. ${\int }_{-1}^{1}x|x|dx$
   (a) $\frac{2}{3}$ (b) 0 (c) 1 (d) 2
6. $\int e^x\left(\frac{1}{x}-\frac{1}{x^2}\right)dx$
   (a) $\frac{e^x}{x^2}$ (b) $x{e}^x$ (c) $e^x\log x$ (d) $\frac{e^x}{x}$
7. $\int \frac{dx}{\sqrt{1-x^2}}$
   (a) ${\sin }^{-1}x$ (b) ${\cos }^{-1}x$ only (c) ${\tan }^{-1}x$ (d) $\log x$
8. ${\int }_0^1\frac{dx}{1+x^2}$
   (a) $\frac{\pi }{2}$ (b) $\pi$ (c) $\frac{\pi }{4}$ (d) 1
9. $\int x{e}^{x}dx$
   (a) $x{e}^x$ (b) $x^2{e}^x$ (c) $e^x(x+1)$ (d) $e^x(x-1)$
10. $\int {\cos }^{2}xdx$
    (a) $\frac{{\sin }^{3}x}{3}$ (b) $\frac{x}{2}+\frac{\sin 2x}{4}$ (c) $\frac{x}{2}-\frac{\sin 2x}{4}$ (d) $2\cos x$
11. ${\int }_0^{\pi }x\sin xdx$
    (a) 0 (b) $2\pi$ (c) $\pi$ (d) $\frac{\pi }{2}$
12. $\int \frac{2x}{x^2+1}dx$
    (a) $\log (x^2+1)$ (b) ${\tan }^{-1}x$ (c) $2\log x$ (d) $\frac{1}{x^2+1}$
13. ${\int }_0^{\pi /2}\frac{{\sin }^{n}x}{{\sin }^{n}x+{\cos }^{n}x}dx$
    (a) $\frac{\pi }{2}$ (b) $\frac{\pi }{4}$ (c) $\pi$ (d) 0
14. $\int \tan xdx$
    (a) ${\sec }^{2}x$ (b) $\log |\cos x|$ (c) $\log |\sec x|$ (d) ${\tan }^{2}x$
15. ${\int }_{-\pi /2}^{\pi /2}\cos xdx$
    (a) 0 (b) $\pi$ (c) 1 (d) 2
16. $\int \frac{dx}{x^2-1}$
    (a) $\frac{1}{2}\log \left|\frac{x-1}{x+1}\right|$ (b) $\log |x^2-1|$ (c) ${\tan }^{-1}x$ (d) $\frac{1}{2}\log \left|\frac{x+1}{x-1}\right|$
17. ${\int }_0^{1}x{e}^{x^2}dx$
    (a) $\frac{e-1}{2}$ (b) $e-1$ (c) $e$ (d) $\frac{e}{2}$
18. $\int \frac{dx}{{\sin }^{2}x}$
    (a) $\cot x$ (b) $-\csc x$ (c) $\tan x$ (d) $-\cot x$
19. ${\int }_0^3|x-2|dx$
    (a) 3 (b) $\frac{5}{2}$ (c) 2 (d) $\frac{3}{2}$
20. $\int \log xdx$
    (a) $\frac{1}{x}$ (b) $x\log x$ (c) $x\log x-x$ (d) $\log x-x$

1 (c) · 2 (a) · 3 (d) · 4 (b) · 5 (b) · 6 (d) · 7 (a) · 8 (c) · 9 (d) · 10 (b) · 11 (c) · 12 (a) · 13 (b) · 14 (c) · 15 (d) · 16 (a) · 17 (a) · 18 (d) · 19 (b) · 20 (c)

Assertion–Reason (6)

(a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.

1. A: ${\int }_{-1}^1{x}^5\cos xdx=0$. R: $x^5\cos x$ is an odd function.
2. A: $\int \frac{dx}{x}=\log x+C$ for all $x\ne 0$. R: $\frac{d}{dx}\log |x|=\frac{1}{x}$.
3. A: ${\int }_0^{\pi /2}\frac{\cos x}{\sin x+\cos x}dx=\frac{\pi }{4}$. R: ${\int }_0^{a}f(x)dx={\int }_0^{a}f(a-x)dx$.
4. A: $\int e^x(\tan x+{\sec }^{2}x)dx=e^x\tan x+C$. R: $\int e^x[f(x)+f^{\prime }(x)]dx=e^{x}f(x)+C$.
5. A: ${\int }_0^2[x]dx=1$. R: $[x]$ is continuous on $[0,2]$.
6. A: $\int {\sin }^{2}xdx=\frac{{\sin }^{3}x}{3}+C$. R: $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$.

Key: 1 (a) · 2 (d) · 3 (a) · 4 (a) · 5 (c) · 6 (d)

Answers to the practice set