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Question Bank No: 1

1. If f(-x) + f(x) = 0, then $\int_{a}^{x}$ $(t)$ dt is

a)		an odd function
b)		an even function
c)		a periodic function
d)		None of these

2. If f is a continuous periodic function with period T, then $\int_{2 T}^{a + 2 T} f (x) dx$

a)		depends upon a
b)		is independent of T
c)		depends upon T
d)		independent of a

3. If the function f(x) = ${Pe}^{2 x}$ + ${Qe}^{x}$ + Rx satisfies the conditions : f(0) = -1, f'(log 2) = 31 and $\int_{0}^{\log 4} (f (x) - Rx) dx = \frac{39}{2}$ ,then

a)		P = 5, Q = -6, R = 3
b)		P = 3, Q = 2, R= 3
c)		P = -5, Q = 6, R = 3
d)		P = -5, Q = 6, R= 3

4. $\int_{0}^{1.5} x [x^{2}] dx ......$

a)		$\frac{1}{2}$
b)		$\frac{3}{4}$
c)		$\frac{3}{2}$
d)		None of these

5. $\int_{- π / 2}^{π / 2} \sin (\log (x + \sqrt{x^{2 + 1})}) dx =$

a)		0
b)		-1
c)		1
d)		None of these

6. The function f(x) = $\int_{0}^{x}$ log $(\frac{1 - x}{1 + x})$ dx is

a)		an even function
b)		an odd function
c)		a periodic function
d)		None of these

7. $\frac{d}{dx} \int_{x}^{x^{2}} \sqrt{cost} dt = .....$

a)		2x cos $x^{2}$ - cos x
b)		2x $\sqrt{\cos x^{2}}$ - cos x
c)		2 x cos $x^{2}$ - $\sqrt{\cos x}$
d)		None of these

8. $\int_{0}^{1} x (1 - x)^{99} dx = .......$

a)		$\frac{1}{10100}$
b)		$\frac{1}{11000}$
c)		$\frac{1}{1010}$
d)		None of these

9. $\int_{- 5}^{5} | x | dx = .......$

a)		6
b)		9
c)		0
d)		None of these

10. $\int_{2}^{4} x \sqrt{6 - x} dx = .......$

a)		$\frac{32}{5} (3 + \sqrt{2})$
b)		$\frac{32}{5} (3 - \sqrt{2})$
c)		$\frac{16}{5} (3 - \sqrt{2})$
d)		None of these

11. The function f(x) = $\int_{o}^{x}$ log $(t + \sqrt{1 + t^{2}})$ dt is

a)		an even function
b)		an odd function
c)		a periodic function
d)		None of these

12. $\int_{- 8}^{8} (\sin^{193} + x + x^{295}) dx$ is equal to

a)		0
b)		2 $(8^{295} + 1)$
c)		$8^{295}$ + 2
d)		None of these

13. If $\int_{0}^{n π + t} | sinx |$ dx = k – cos t (0 $\leq$ t $\leq$ $π$ ), then the value of k is

a)		2n
b)		4n + 1
c)		3n + 1
d)		2n + 1

14. Value of $\int_{- 1}^{1} \frac{dx}{- 1 \sqrt{1 - 2 ax + a^{2}}},$ (-1 < $α$ $<$ 1) is

a)		2
b)		$\frac{2}{α}$
c)		- $\frac{2}{α}$
d)		None of these

15. Value of $\int_{1}^{2}$ xlog (1 $x^{- 1})$ dx is

a)		log $(\frac{27}{2} e^{- 1 / 2})$
b)		$\frac{1}{2} \log (\frac{27 e}{16})$
c)		log2-1+ $\frac{π}{2}$
d)		$\frac{1}{2} (\log (\frac{2}{e}) + \frac{π}{2})$

16. $\int_{0}^{π}$ x log sin x dx

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

17. Value of $\int_{0}^{2 a}$ $\frac{f (x)}{f (x) + f (2 a - x)}$ dx is

a)		0
b)		$\frac{π}{4}$
c)		a
d)		None of these

18. $\int_{0}^{100} e^{x - [- x]} dx =$

a)		100(e – 1)
b)		100e
c)		100(e + 1)
d)		None of these

19. Value of $\int_{0}^{π}$ $\sin^{n} x$ $\cos^{2 m + 1} x dx$ (where m, n $\in$ N) is

a)		$\frac{(2 m + 1)!}{n!}$
b)		$\frac{(2 m + 1)!}{n^{2}}$
c)		$\int_{0}^{π} \cos^{2 m + 1} xdx$
d)		None of these

20. Value of $\int_{o}^{n}$ $[x]$ dx (where n $\in$ N) is

a)		$\frac{n (n + 1)}{2}$
b)		$\frac{n (n - 1)}{2}$
c)		n(n – 1)
d)		None of these

21. $\int_{0}^{\frac{π}{4}}$ $\frac{\sqrt{\tan x}}{\sin x \cos x}$ dx equals

a)		1
b)		2
c)		0
d)		4

22. I = $\int_{0}^{π / 2}$ $\frac{\sin 2 x}{1 + 4 \cos^{2} x}$ dx is equal to

a)		$\frac{1}{4}$ log 2
b)		$\frac{1}{4}$ log 4
c)		$\frac{1}{4}$ log 3
d)		$\frac{1}{4}$ log 5

23. The function F(x) = $\int_{0}^{x}$ $\log [t + \sqrt{1 + t^{2}}]$ dt is

a)		an even function
b)		an odd function
c)		a periodic function
d)		none of these

24. $\int_{- 1}^{1}$ $x | x |$ dx is

a)		2
b)		1
c)		0
d)		none of these

25. The value of the integral $\int_{0}^{2 a}$ $\frac{f (x)}{f (x) + f (2 a - x)}$ dx is

a)		a
b)		2a
c)		1
d)		0

26. $\int_{0}^{π / 2}$ $\frac{\sqrt{\cot x}}{\sqrt{\cot x} + \sqrt{\tan x}}$ dx is

a)		$\frac{π}{4}$
b)		$\frac{π}{2}$
c)		$π$
d)		0

27. $\int_{1}^{2}$ $e^{- 1 / x} \frac{dx}{x^{2}}$ is equal to

a)		$\frac{1}{\sqrt{e}} + \frac{1}{e}$
b)		$\frac{1}{\sqrt{e}} - \frac{1}{e}$
c)		$\frac{1}{e} - \frac{1}{\sqrt{e}}$
d)		none of these

28. $\int_{1}^{3}$ $\frac{\cos (\log x)}{x}$ dx is equal to

a)		log (sin 3)
b)		log 3
c)		sin (log 3)
d)		sin (log 2)

29. $\int_{- 8}^{8}$ $(a x^{2} + bx + c) dx$ is a function of

a)		c
b)		a,b
c)		a,b,c
d)		b,c

30. $\int_{- π / 2}^{π / 2}$ $\sin x dx$ equals

a)		0
b)		1
c)		-1
d)		2

31. $\int_{1}^{e} \frac{1 + \log x}{2 x} dx$ is equal to

a)		$\frac{1}{4}$
b)		e
c)		$\frac{3}{4}$
d)		$\frac{1}{e}$

32. $\int_{3}^{6}$ $2 [x] dx$ is equal to

a)		12
b)		30
c)		18
d)		24

33. $\int_{π / 6}^{π / 3}$ $\frac{dx}{\sin 2 x}$ is equal to

a)		$\frac{1}{2}$ log (-1)
b)		log (-1)
c)		log 3
d)		log $\sqrt{3}$

34. $\int_{4}^{5}$ $(| x - 4 | + | x - 5 |) dx$ =

a)		1
b)		2
c)		0
d)		4

35. $\int_{3}^{4}$ $\sqrt{(x - 3) (4 - x)}$ dx =

a)		$\frac{π}{2}$
b)		$\frac{π}{4}$
c)		$\frac{π}{8}$
d)		0

36. $\int_{4}^{5} \frac{dx}{\sqrt{(x - 4) (5 - x)}} =$

a)		$\frac{π}{2}$
b)		$π$
c)		2 $π$
d)		$\frac{3 π}{2}$

37. $\int_{0}^{1} x^{2} e^{x} dx$

a)		$e - 2$
b)		$e + 2$
c)		$e^{2} - 2$
d)		$e^{2} + 2$