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Question Bank No: 1

1. $\int \frac{\cos 2 x}{{(sinx + cosx)}^{2}} dx =$

a)		$\frac{- 1}{sinx + cosx} + c$
b)		log (sin x + cos x) + C
c)		log (sin x – cos x) + C
d)		log ${(sinx + cosx)}^{2} + c$

2. $\int e^{xloga} . e^{x} dx =$

a)		${(ae)}^{x}$
b)		$\frac{{(ae)}^{x}}{\log (ae)}$
c)		$\frac{e^{x}}{1 + loga}$
d)		None of these

3. $\int {| x |}^{3} dx =$

a)		$\frac{x^{4}}{4}$
b)		$- \frac{x^{4}}{4}$
c)		$\frac{{\| x \|}^{4}}{4}$
d)		None of these

4. $\int \frac{xdx}{1 + 4^{x}} =$

a)		$\tan^{- 1} x^{2} + c$
b)		$\frac{1}{2} \tan^{- 1} x^{2} + c$
c)		log (1 + $x^{4}$ ) + C
d)		None of these

5. If $\int \frac{2^{x}}{\sqrt{1 - 4^{x}}} dx = K \sin^{- 1} (2 x)$ , then k is

a)		log 2
b)		$\frac{1}{2}$ log 2
c)		$\frac{1}{2}$
d)		${(\log 2)}^{- 1}$

6. $\int \frac{(sinx)^{3 / 2}}{(cosx)^{7 / 2}} dx =$

a)		$\frac{3}{2} (tanx)^{1 / 2}$ + C
b)		$\frac{2}{3} (tanx)^{3 / 2}$ + C
c)		$\frac{2}{5} (tanx)^{5 / 2}$ + C
d)		None of these

7. $\int \frac{\log (x + \sqrt{x^{2} + 4}}{\sqrt{x^{2} + 4}} dx =$

a)		$\frac{1}{\sqrt{x^{2} + 4}} + c$
b)		$\frac{1}{2}$ $[\log (x + \sqrt{x^{2} + 4]^{2}} + c$
c)		2 $(x^{2} + 4)^{1 / 2}$ + C
d)		None of these

8. $\int$ (log(logx) + ${(logx)}^{- 1}$ )dx=

a)		x log (log x) + c
b)		x log (log x) + $\frac{x}{logx}$
c)		x log (log x) – $\frac{x}{logx}$ + C
d)		None of these

9. $\int \frac{cosx}{\cos (x - a)} dx$ (f)cosxcos(x-a)dx

a)		x cos a + sin a log $\| \cos (x - a) \|$ +C
b)		x cos a - sin a log $\| \cos (x - a) \|$ +C
c)		sin a - sin a log $\| \cos (x - a) \|$ +C
d)		None of these

10. $\int \frac{\sin 2 xdx}{3 + 4 \sin^{2} x} =$

a)		4 log (3 + 4 $\sin^{2}$ x) + C
b)		$\frac{1}{2} \tan^{- 1} (\frac{2 sinx}{\sqrt{3}}) + c$
c)		$\frac{1}{4} \log (3 + 4 \sin^{2} x) + c$
d)		None of these

11. $\int | x | dx$

a)		$\frac{1}{2} x^{2}$
b)		$- \frac{x^{2}}{2}$
c)		0
d)		$\frac{1}{2} x \| x \|$

12. $\int \frac{1}{f (x)} dx = \log (f (x))^{2} + c,$ then f(x) is

a)		x + $α$
b)		2x + $α$
c)		$\frac{x}{2}$ + $α$
d)		$x^{2}$ + $α$

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

a)		log ( $e^{2 x}$ + 1)
b)		log ( $e^{x} + e^{- x}$ )
c)		$\tan^{- 1} (e^{x})$
d)		None of these

14. $\int {(\frac{\tan \frac{1}{x}}{x})}^{2} dx =$

a)		x– tan x + C
b)		$\frac{1}{x} - \tan (\frac{1}{x}) + c$
c)		$\frac{1}{x}$ + tan $(\frac{1}{x})$ + c
d)		None of these

15. $\int$ $\frac{d (x^{2} + 1)}{\sqrt{x^{2} + 2}}$ =

a)		2 $\sqrt{x^{2} + 2 + c}$
b)		$\sqrt{x^{2} + 2 + c}$
c)		$\frac{1}{(x^{2} + 2)^{3 / 2}} + c$
d)		None of these

16. $\int e^{x} {(\frac{1 - x}{1 + x^{2}})}^{2} dx$ =

a)		$\frac{e^{x}}{1 + x^{2}}$ +c
b)		$\frac{2 x e^{x}}{(1 + x^{2})^{2}} + c$
c)		$\frac{e^{- x}}{1 + x^{2}}$ +c
d)		None of these

17. Anti-derivative of $\frac{x}{\cos^{2} x}$ is

a)		x tan x
b)		x tan x + log (cos x)
c)		log (cos x)
d)		cot x

18. Integral of $\sqrt{1 + x^{2}}$ w.r.t. $x^{2}$ is

a)		$\frac{2}{3} \frac{(1 + x^{2})^{3 / 2}}{x} + c$
b)		$\frac{2}{3} (1 + x^{2})^{3 / 2} + c$
c)		$\frac{2}{3} x (1 + x^{2})^{3 / 2} + c$
d)		None of these

19. If f(x) = $\int$ $\cot^{4}$ + $\frac{1}{3}$ $\cot^{3}$ x-cotx and $f (\frac{π}{2})$ = $\frac{π}{2}$ , then f(x) is

a)		$π$ – x
b)		x – $π$
c)		x
d)		None of these

20. $\int$ $e^{- logx}$ dx=

a)		$-$ $e^{- logx}$ + C
b)		$- x$ $e^{- logx}$ + C
c)		log $\| x \|$ + C
d)		x $e^{- logx}$ + C

21. Integral of f (x) = $\sqrt{1 + x^{2}}$ with respect to $x^{2}$ is

a)		$\frac{2}{3} \frac{{(1 + x^{2})}^{\frac{3}{2}}}{x} + K$
b)		$\frac{2}{3} {(1 + x^{2})}^{3 / 2} + C$
c)		$\frac{2}{3} {x (1 + x^{2})}^{3 / 2} + C$
d)		none of these

22. $\int \frac{\log (x + 1) - \log x}{x (x + 1)} dx$ equal to

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

23. The value $\int \sec^{3} x dx$ will be

a)		$\frac{1}{2}$ [sec x tan x + log (sec x +tan x)]
b)		$\frac{1}{3}$ [sec x tan x + log (sec x +tan x)]
c)		$\frac{1}{4}$ [sec x tan x + log (sec x +tan x)]
d)		$\frac{1}{8}$ [sec x tan x + log (sec x +tan x)]

24. $\int \frac{x^{2} + 1}{x (x^{2} - 1)}$ dx is equal to

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

25. $\int (e^{a \log x} + e^{x \log a})$ dx is equal to

a)		$\frac{x^{a + 1}}{a + 1} + c$
b)		$\frac{x^{a + 1}}{a + 1} + \frac{a^{x}}{\log a} + c$
c)		$x^{a + 1} + a^{x} + c$
d)		$\frac{x^{a + 1}}{a + 1} + \frac{a^{x}}{\log a}$

26. $\int \sqrt{1 + \sin x} \cdot f (x) dx = \frac{2}{3} {(1 + \sin x)}^{3 / 2} + C$ then f (x) =

a)		cos x
b)		sin x
c)		tan x
d)		1

27. $\int \frac{\cot x}{\sqrt{\sin x}} dx =$

a)		$\frac{- 2}{\sqrt{\sin x}} + c$
b)		$\frac{2}{\sqrt{\sin x}} + c$
c)		$2 \sqrt{\sin x} + c$
d)		$\frac{1}{2 \sqrt{\sin x}} + c$

28. $\int \frac{dx}{x + \sqrt{x}}$ =

a)		$\log_{e} (\sqrt{x} + 1) + c$
b)		2 $\log_{e} (1 - \sqrt{x}) + c$
c)		2 $\log (\sqrt{x} + 1) + c$
d)		2 $\log_{e} (\sqrt{x} - 1) + c$

29. $\int \frac{a^{\sqrt{x}}}{\sqrt{x}} dx$ equals to

a)		${2 a}^{\sqrt{x}} \log_{e} a$
b)		${2 a}^{\sqrt{x}} \log_{a} e$
c)		${2 a}^{\sqrt{x}} \log_{10} a$
d)		${2 a}^{\sqrt{x}} \log_{a} 10$

30. If $\frac{d}{dx} (f (x))$ = g (x) then $\int f (x) g (x) dx =$

a)		f(x)
b)		$\frac{{[f (x)]}^{2}}{2}$
c)		g(x)
d)		$\frac{{(g (x))}^{2}}{2}$

31. $\int \frac{\sqrt{tanx}}{2 \sin x \cos x} dx =$

a)		$\sqrt{tanx}$
b)		2 $\sqrt{tanx}$
c)		$\frac{1}{2} \sqrt{tanx}$
d)		$\frac{1}{2}$ tan x

32. $\int \frac{e^{x} (1 + sinx)}{1 + \cos x} dx =$

a)		log tan x
b)		$e^{x} \tan \frac{x}{2}$
c)		sin (log x)
d)		$e^{x} \cot x$

33. If $\int \frac{2^{x}}{\sqrt{1 - 4^{x}}}$ dx=K $\sin^{- 1} (2 x)$ then K =

a)		log 2
b)		$\frac{1}{2}$
c)		$\frac{1}{2}$ log 2
d)		$\frac{1}{\log 2}$

34. $\int e^{\tan^{- 1} x} [1 + \frac{x}{1 + x^{2}}] dx is$

a)		x $e^{\tan^{- 1} x}$
b)		$\frac{1}{2} x e^{\tan^{- 1} x}$
c)		$e^{\tan^{- 1} x}$
d)		$\frac{1}{2} e^{\tan^{- 1} x}$

35. $\int x^{x} (1 + logx) dx =$

a)		$\frac{{(1 + logx)}^{2}}{2}$
b)		$x^{2} logx$
c)		$x^{2} x$
d)		$x^{2}$