Integrals MCQ Set-1

Question 1

$\int \frac{1}{2x-1} dx$

Accepted Answer

$\frac{1}{2} \log|2x-1| + C$

Answer

$\log|2x-1| + C$

Answer

$-\frac{1}{2} \log|2x-1| + C$

Answer

$2 \log|2x-1| + C$

Question 2

$\int (3\sin^2 x + \cos^2 x) dx$

Accepted Answer

$2x – \frac{1}{2}\sin 2x + C$

Answer

$x + \sin 2x + C$

Answer

$2x + \frac{1}{2}\cos 2x + C$

Answer

$x – \cos 2x + C$

Question 3

$\int e^{3-2x} dx$

Accepted Answer

$-\frac{1}{2} e^{3-2x} + C$

Answer

$e^{3-2x} + C$

Answer

$\frac{1}{2} e^{3-2x} + C$

Answer

$-\frac{1}{3} e^{3-2x} + C$

Question 4

$\int \sin(e^x) e^x dx$

Accepted Answer

$-\cos(e^x) + C$

Answer

$\cos(e^x) + C$

Answer

$\sin(e^x) + C$

Answer

$e^x \cos(e^x) + C$

Question 5

$\int e^{\log x^{-2}} dx$

Accepted Answer

$-\frac{1}{x} + C$

Answer

$\frac{1}{x} + C$

Answer

$\frac{x^{-3}}{-3} + C$

Answer

$\frac{x^2}{2} + C$

Question 6

$\int (\sec^2 x - \csc^2 x) dx$

Accepted Answer

$	an x + \cot x + C$

Answer

$	an x - \cot x + C$

Answer

$	an x + 2\cot x + C$

Answer

$2	an x + \cot x + C$

Question 7

$\int \frac{\sin 2x}{1 + \cos 2x} dx$

Accepted Answer

$-\frac{1}{2} \log|\cos x| + C$

Answer

$-\log|\cos x| + C$

Answer

$\log|\sec x| + C$

Answer

$\frac{1}{2} \log|\sec x| + C$

Question 8

$\int \frac{\cos 2x - 1}{\cos 2x + 1} dx$

Accepted Answer

$x - 	an x + C$

Answer

$	an x - x + C$

Answer

$\cot x - x + C$

Answer

$x - \cot x + C$

Question 9

$\int \frac{\cos 2x}{\cos x} dx$

Accepted Answer

$2\sin x - \log|\sec x + 	an x| + C$

Answer

$\sin x - \log|\sec x + 	an x| + C$

Answer

$2\sin x + \log|\sec x + 	an x| + C$

Answer

$2\cos x - \log|\sec x + 	an x| + C$

Question 10

$\int \frac{1}{(3-x)^2} dx$

Accepted Answer

$\frac{1}{3-x} + C$

Answer

$-\frac{1}{3-x} + C$

Answer

$-\log|3-x| + C$

Answer

$\frac{-1}{(3-x)^3} + C$

Question 11

$\int \sin^{-1} (\cos x) dx$

Accepted Answer

$\frac{\pi x}{2} - \frac{x^2}{2} + C$

Answer

$\frac{\pi x}{2} + \frac{x^2}{2} + C$

Answer

$\frac{\pi}{2} - x + C$

Answer

$\frac{\pi x}{2} - x^2 + C$

Question 12

$\int e^x \left(\frac{x-1}{x^2}\right) dx$

Accepted Answer

$\frac{e^x}{x} + C$

Answer

$\frac{e^x}{x^2} + C$

Answer

$e^x \log|x| + C$

Answer

$-\frac{e^x}{x} + C$

Question 13

$\int \frac{dx}{\sqrt{1-4^x}}$

Accepted Answer

$\frac{1}{\log 2} \sin^{-1}(2^x) + C$

Answer

$\sin^{-1}(2^x) + C$

Answer

$\frac{1}{\log 4} \sin^{-1}(2^x) + C$

Answer

$\frac{2}{\log 2} \sin^{-1}(2^x) + C$

Question 14

$\int e^{	an^{-1} \sqrt{x}} \frac{1}{\sqrt{x} + x \sqrt{x}} dx$

Accepted Answer

$2 e^{	an^{-1} \sqrt{x}} + C$

Answer

$e^{	an^{-1} \sqrt{x}} + C$

Answer

$\frac{1}{2} e^{	an^{-1} \sqrt{x}} + C$

Answer

$e^{	an^{-1} x} + C$

Question 15

$\int \frac{e^{x}+e x^{e-1}}{e^{x}+x^{e}} d x$

Accepted Answer

$\log|e^x + x^e| + C$

Answer

$\frac{1}{e} \log|e^x + x^e| + C$

Answer

$e \log|e^x + x^e| + C$

Answer

$\log|e^x + x^{e-1}| + C$

Question 16

$\int \frac{1}{1 + e^x} dx$

Accepted Answer

$x - \log|e^x + 1| + C$

Answer

$\log|1+e^x| + C$

Answer

$\log|e^x + 1| - x + C$

Answer

$\frac{1}{2} \log|1+e^x| + C$

Question 17

If $\int \frac{1+x^2}{1+x^3} dx = \log|x| + f(x) + C$, then $f'(x)$ is:

Accepted Answer

$\frac{x^4}{x+x^5}$

Answer

$\frac{1}{x+x^5}$

Answer

$x^5$

Answer

$\frac{x^3}{1+x^5}$

Question 18

$\int 	an 3x dx$

Accepted Answer

$\frac{1}{3} \log|\sec 3x| + C$

Answer

$3 \log|\sec 3x| + C$

Answer

$\log|\sec 3x| + C$

Answer

$-\frac{1}{3} \log|\cos 3x| + C$

Question 19

$\int a^{\sqrt{x}} \frac{1}{\sqrt{x}} dx$

Accepted Answer

$\frac{2 a^{\sqrt{x}}}{\log a} + C$

Answer

$\frac{a^{\sqrt{x}}}{\log a} + C$

Answer

$2 a^{\sqrt{x}} \log a + C$

Answer

$a^{\sqrt{x}} \frac{\sqrt{x}}{2} + C$

Question 20

$\int \cot^2 x \csc^2 x dx$

Accepted Answer

$-\frac{1}{3} \cot^3 x + C$

Answer

$\frac{1}{3} \cot^3 x + C$

Answer

$\cot^3 x + C$

Answer

$\frac{1}{2} \cot^2 x + C$

Question 21

$\int 	an^6 x \sec^2 x dx$

Accepted Answer

$\frac{1}{7} 	an^7 x + C$

Answer

$\frac{1}{6} 	an^6 x + C$

Answer

$	an^7 x + C$

Answer

$\frac{1}{7} \sec^7 x + C$

Question 22

$\int \frac{x}{1+x^4} dx$

Accepted Answer

$\frac{1}{2} 	an^{-1}(x^2) + C$

Answer

$	an^{-1}(x^2) + C$

Answer

$\log|1+x^4| + C$

Answer

$\frac{x^2}{1+x^4} + C$

Question 23

$\int \frac{\cot x}{\sqrt{\sin x \cos x}} dx$

Accepted Answer

$-2 \sqrt{\sin x} + C$

Answer

$-2 \sqrt{\cos x} + C$

Answer

$2 \sqrt{\sin x} + C$

Question 24

$\int \frac{\cos x}{a+b \sin x} dx$

Accepted Answer

$\frac{1}{b} \log|a+b \sin x| + C$

Answer

$b \log|a+b \sin x| + C$

Answer

$\frac{1}{a} \log|a+b \sin x| + C$

Answer

$\log|a+b \sin x| + C$

Question 25

$\int \frac{1}{e^{-x} + 1} dx$

Accepted Answer

$\log|1+e^x| + C$

Answer

$x + \log|1+e^x| + C$

Answer

$x - \log|1+e^x| + C$

Answer

$\frac{1}{2} \log|1+e^x| + C$

Question 26

$\int \frac{\cos^3 x}{\sin^2 x + \sin x} dx$

Accepted Answer

$\log|\sin x| - \sin x + C$

Answer

$\log|\sin x| - \cos x + C$

Answer

$\log|\sin x| - \sin x + \cos x + C$

Answer

$\log|\sin x| + \sin x + C$

Question 27

$\int \frac{\cos x - \sin x}{\sin x + \cos x} dx$

Accepted Answer

$\log|\sin x + \cos x| + C$

Answer

$-\log|\sin x + \cos x| + C$

Answer

$\frac{1}{2} \log|\sin x + \cos x| + C$

Answer

$\sin x + \cos x + C$

Question 28

$\int e^x (\cot x - \csc^2 x + 1) dx$

Accepted Answer

$e^x (1 + \cot x) + C$

Answer

$e^x (\cot x) + C$

Answer

$e^x (\cot x - 1) + C$

Answer

$e^x (\csc^2 x) + C$

Question 29

$\int \frac{1-x}{\sqrt{x}} dx$

Accepted Answer

$2\sqrt{x} - \frac{2}{3} x^{3/2} + C$

Answer

$\sqrt{x} - x^{3/2} + C$

Answer

$\frac{1}{2\sqrt{x}} - \frac{1}{2\sqrt{x}} + C$

Answer

$x^{1/2} - x^{3/2} + C$

Question 30

$\int x \sin x dx$

Accepted Answer

$-x \cos x + \sin x + C$

Answer

$-x \cos x - \sin x + C$

Answer

$x \cos x - \sin x + C$

Answer

$x \sin x + \cos x + C$

Question 31

$\int \frac{\cos 2x}{(\sin x + \cos x)^2} dx$

Accepted Answer

$\log_e|\sin x + \cos x| + C$

Answer

$\frac{-1}{\sin x + \cos x} + C$

Answer

$\log_e|\sin x - \cos x| + C$

Answer

$\frac{1}{(\sin x + \cos 	heta)^2} + C$

Question 32

$\int \frac{dx}{\sin (x-a) \sin (x-b)}$

Accepted Answer

$\csc (b-a) \log |\frac{\sin (x-b)}{\sin (x-a)}| + C$

Answer

$\sin (b-a) \log |\frac{\sin (x-b)}{\sin (x-a)}| + C$

Answer

$\csc (b-a) \log |\frac{\sin (x-a)}{\sin (x-b)}| + C$

Answer

$\sin (b-a) \log |\frac{\sin (x-a)}{\sin (x-b)}| + C$

Question 33

$\int 	an^{-1} \sqrt{x} dx$

Accepted Answer

$(x + 1)	an^{-1}\sqrt{x} - \sqrt{x} + C$

Answer

$x 	an^{-1}\sqrt{x} - \sqrt{x} + C$

Answer

$\sqrt{x} - x 	an^{-1}\sqrt{x} + C$

Answer

$\sqrt{x} - (x + 1)	an^{-1}\sqrt{x} + C$

Question 34

$\int \frac{\sin^6 x}{\cos^8 x} dx$

Accepted Answer

$\frac{1}{7} 	an^7 x + C$

Answer

$\frac{1}{5} 	an^6 x + C$

Answer

$\frac{1}{5} 	an^7 x + C$

Answer

None of these

Question 35

$\int \frac{x}{(x-1)(x-2)} dx$

Accepted Answer

$\log_e|\frac{(x-2)^2}{x-1}| + C$

Answer

$\log_e|\frac{(x-1)^2}{x-2}| + C$

Answer

$\log_e|(\frac{x-1}{x-2})^2| + C$

Answer

$\log_e|\frac{x-1}{x-2}| + C$

Question 36

$\int (x^2 + 2x + 2)^{-1} dx$

Accepted Answer

$	an^{-1} (x + 1) + C$

Answer

$x 	an^{-1} (x + 1) + C$

Answer

$(x + 1) 	an^{-1}x + C$

Answer

$x 	an^{-1}x + C$

Question 37

$\int_{0}^{\pi/2} \sqrt{1 - \sin 2x} dx$

Accepted Answer

$2(\sqrt{2} - 1)$

Answer

$2\sqrt{2}$

Answer

$2(\sqrt{2} + 1)$

Answer

2

Question 38

The anti-derivative of $(\sqrt{x} + 1/\sqrt{x})$ is

Accepted Answer

$\frac{2}{3} x^{3/2} + 2x^{1/2} + C$

Answer

$\frac{1}{3} x^{1/3} + 2x^{1/2} + C$

Answer

$\frac{2}{3} x^{2/3} + \frac{1}{2} x^2 + C$

Answer

$\frac{3}{2} x^{3/2} + \frac{1}{2} x^{1/2} + C$

Question 39

If $f’(x) = 4x^3 - 3/x^4$ such that $f(2) = 0$. Then $f(x)$ is

Accepted Answer

$x^4 + x^{-3} - \frac{129}{8}$

Answer

$x^3 + x^{-4} + \frac{129}{8}$

Answer

$x^4 + x^{-3} + \frac{129}{8}$

Answer

$x^3 + x^{-4} - \frac{129}{8}$

Question 40

$\int_{1}^{2} x^{2} d x$

Accepted Answer

$\frac{7}{3}$

Answer

1

Answer

$\frac{1}{3}$

Answer

0

Question 41

$\int_{0}^{2}\left(x^{2}+3\right) d x$

Accepted Answer

$\frac{26}{3}$

Answer

$\frac{25}{3}$

Answer

$\frac{24}{3}$

Answer

None of these

Question 42

Evaluate: $\int_{0}^{\pi / 4} \sqrt{1-\sin 2 x} d x$

Accepted Answer

$\sqrt{2} – 1$

Answer

$\sqrt{2} + 1$

Answer

$\sqrt{2}$

Answer

None of these

Question 43

Evaluate: $\int_{0}^{2 \pi} \sin \left(\frac{\pi}{4}+\frac{x}{2}\right) d x$

Accepted Answer

$2\sqrt{2}$

Answer

$-2\sqrt{2}$

Answer

$-2$

Answer

$\sqrt{2}$

Question 44

Evaluate: $\int_{1}^{2} \frac{d x}{x^{2}}$

Accepted Answer

$\frac{1}{2}$

Answer

1

Answer

2

Answer

-1

Question 45

$\int (2	an x – 3\cot x)^2 dx$

Accepted Answer

$4	an x – 9\cot x – 25x + C$

Answer

$-4	an x – 9\cot x – 25x + C$

Answer

$-4	an x + 9 \cot x + 25x + C$

Answer

$4	an x + 9\cot x + 25x + C$

Question 46

$\int e^x (\cos x - \sin x) dx$

Accepted Answer

$e^x \cos x + c$

Answer

$e^x \cos x - c$

Answer

$-e^x \cos x - c$

Answer

$-e^x \cos x + c$

Question 47

$\int_{-\pi/2}^{\pi/2} (x^3 + x \cos x + 	an^5 x + 1) dx$

Accepted Answer

$\pi$

Answer

0

Answer

1

Answer

$2\pi$

Question 48

If $f(a+b-x) = f(x)$, then $\int_{a}^{b} x f(x) d x$ is equal to:

Accepted Answer

$\frac{a+b}{2} \int_{a}^{b} f(x) d x$

Answer

$\frac{a-b}{2} \int_{a}^{b} f(x) d x$

Answer

$(a+b) \int_{a}^{b} f(x) d x$

Answer

$\frac{a}{2} \int_{a}^{b} f(x) d x$

Question 49

If $f(x) = \int_{0}^{x} t \sin t dt$, then $f'(x)$ is:

Accepted Answer

$x \sin x$

Answer

$\sin x$

Answer

$t \sin t$

Answer

$x \cos x$

Question 50

$\int_{-\pi/4}^{\pi/4} \frac{1}{1 + \cos 2x} dx$

Accepted Answer

1

Answer

0

Answer

$\frac{1}{2}$

Answer

2

Question 51

$\int_{0}^{2/3} \frac{1}{4 + 9x^2} dx$

Accepted Answer

$\frac{\pi}{24}$

Answer

$\frac{\pi}{12}$

Answer

$\frac{\pi}{6}$

Answer

$\frac{\pi}{36}$

Question 52

$\int \frac{10x^9 + 10^x \log_e 10}{x^{10} + 10^x} dx$

Accepted Answer

$\log |x^{10} + 10^x| + C$

Answer

$\log |10^x| + C$

Answer

$\frac{1}{2} \log |x^{10} + 10^x| + C$

Answer

$e^x (x^{10} + 10^x) + C$

Question 53

$\int_{0}^{\pi/2} \log \left( \frac{4+3\sin x}{4+3\cos x} \right) dx$

Accepted Answer

0

Answer

1

Answer

$\frac{\pi}{2}$

Answer

$\log 3$

Question 54

$\int_{0}^{1} 	an^{-1} \left( \frac{2x-1}{1+x-x^2} ight) dx$

Accepted Answer

0

Answer

$\pi/4$

Answer

1

Answer

$\pi/2$

Question 55

$\int 	an^{-1} \sqrt{x} dx$

Accepted Answer

$x 	an^{-1} \sqrt{x} - \sqrt{x} + 	an^{-1} \sqrt{x} + C$

Answer

$x 	an^{-1} \sqrt{x} + \sqrt{x} + 	an^{-1} \sqrt{x} + C$

Answer

$\sqrt{x} 	an^{-1} \sqrt{x} + x + C$

Answer

$\frac{1}{\sqrt{x}} 	an^{-1} \sqrt{x} + C$

Question 56

$\int e^x \left(\frac{x+3}{(x+4)^2}\right) dx$

Accepted Answer

$e^x \frac{1}{x+4} + C$

Answer

$e^x \frac{x+3}{x+4} + C$

Answer

$e^x \log|x+4| + C$

Answer

$e^x \frac{x+4}{x+3} + C$

Question 57

$\int \frac{x^3}{x+1} dx$

Accepted Answer

$x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\log |1+x|+C$

Answer

$\frac{x^4}{4} - \frac{x^3}{3} + x^2 - x + \log|1+x| + C$

Answer

$x^2 - x + \log|1+x| + C$

Answer

$\frac{x^4}{4} + \log|1+x| + C$

Question 58

The value of $\int e^{x}\left(\frac{x^{2}+2 x+1}{\left(1+x^{2}\right)^{2}}\right) d x$ is:

Accepted Answer

$\frac{e^{x}}{1+x^{2}}+C$

Answer

$\frac{e^{x}}{x^{2}-1}+C$

Answer

$\frac{e^{x}}{x+1}+C$

Answer

$\frac{e^{x}}{x-1}+C$

Question 59

$\int e^x \left(\frac{x+2}{x+4}\right) dx$

Accepted Answer

$e^{x}\left(\frac{x+2}{x+4}\right)+C$

Answer

$e^{x}\left(\frac{x+1}{x+4}\right)+C$

Answer

$e^{x}\left(\frac{x+3}{x+4}\right)+C$

Answer

$e^{x}\left(\frac{x+4}{x+2}\right)+C$

Question 60

$\int \frac{\cos 2x - \cos 2	heta}{\cos x - \cos 	heta} dx$

Accepted Answer

$2(\sin x + x \cos 	heta) + C$

Answer

$2(\sin x + 2x \cos 	heta) + C$

Answer

$2(\sin x - x \cos 	heta) + C$

Answer

$\sin x - x \cos 	heta + C$

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