ഽ\(\frac { 1 }{ { x }^{ 3 } } \)dx is _______.

(a)

\(\frac { -3 }{ { x }^{ 2 } } +c\)

(b)

\(\frac { -1 }{ 2{ x }^{ 2 } } +c\)

(c)

\(\frac { -1 }{ { 3x }^{ 2 } } +c\)

(d)

\(\frac { -2 }{ { x }^{ 2 } } +c\)

ഽ\(\frac { { e }^{ x } }{ \sqrt { { 1+e }^{ x } } } \) dx is _______.

(a)

\(\frac { { e }^{ x } }{ \sqrt { { 1+e }^{ x } } } +c\)

(b)

\(2\sqrt { 1+{ e }^{ x } } +c\)

(c)

\(\sqrt { 1+{ e }^{ x } } +c\)

(d)

\({ e }^{ x }\sqrt { 1+{ e }^{ x } } +c\)

ഽ\(\frac { { e }^{ x } }{ { e }^{ x }+1 } \) dx is _______.

(a)

log\(\left| \frac { { e }^{ x } }{ { e }^{ x }+1 } \right| +c\)

(b)

log\(\left| \frac { { e }^{ x }+1 }{ { e }^{ x } } \right| +c\)

(c)

log\(\left| { e }^{ x } \right| +c\)

(d)

log\(\left| { e }^{ x }+1 \right| +c\)

\(\int _{ 2 }^{ 4 }{ \frac { dx }{ x } } \) is _______.

(a)

log 4

(b)

0

(c)

log 2

(d)

log 8

The value of \(\int _{ -\frac{\pi}{2}}^{ \frac{\pi}{2}}\) cos x dx is _______.

(a)

0

(b)

2

(c)

1

(d)

4

If \(\int _{ 0 }^{ 1 }{ f(x) } dx=1,\int _{ 0 }^{ 1 }{ xf(x) } dx=a\) and \(\int _{ 0 }^{ 1 }{ { x }^{ 2 }f(x) } dx={ a }^{ 2 }\), then \(\int _{ 0 }^{ 1 }{ { (a-x) }^{ 2 } } f(x)\) dx is _______.

(a)

4a²

(b)

0

(c)

2a²

(d)

1

\(\int _{ 0 }^{ \infty }{ { x }^{ 4 }{ e }^{ -x } } \)dx is _______.

(a)

12

(b)

4

(c)

4!

(d)

64

\(\int { \left( x-1 \right) } { e }^{ -x }\) dx = __________ +c

(a)

-xe^x

(b)

xe^x

(c)

-xe^-x

(d)

xe^-x

If \(\int { \frac { { 2 }^{ \frac { 1 }{ x } } }{ { x }^{ 2 } } } dx=k{ 2 }^{ \frac { 1 }{ x } }\) +c, then k is ___________

(a)

\(-\frac { 1 }{ { log }_{ e }2 } \)

(b)

- log_e2

(c)

-1

(d)

\(\frac { 1 }{ 2 } \)

\(\int { { \left| x \right| }^{ 3 } } \)dx = ________________ +c

(a)

\(\frac { { -x }^{ 4 } }{ 4 } \)

(b)

\(\frac { { \left| x \right| }^{ 4 } }{ 4 } \)

(c)

\(\frac { { x }^{ 4 } }{ 4 } \)

(d)

none of these

\(\int { \frac { 2 }{ { \left( { e }^{ x }+{ e }^{ -x } \right) }^{ 2 } } } \) dx = ____________ +c 

(a)

\(\frac { { -e }^{ -x } }{ { e }^{ x }+{ e }^{ -x } } \)

(b)

\(-\frac { { -e }^{ -x } }{ { e }^{ x }+{ e }^{ -x } } \)

(c)

\(\frac { 1 }{ { \left( { e }^{ x }+1 \right) }^{ 2 } } \)

(d)

\(\frac { 1 }{ { e }^{ x }-{ e }^{ -x } } \)

∫ e^{3 log x} (x⁴ +1)^-1 dx = ____________ +c

(a)

\(\log { \left| { x }^{ 4 }+1 \right| } \)

(b)

\(4\log { \left| { x }^{ 4 }+1 \right| } \)

(c)

-4 log |x⁴ +1|

(d)

\(\frac { 1 }{ 4 } \log { \left| { x }^{ 4 }+1 \right| } \)

\(\int _{ -\frac { \pi }{ 2 } }^{ \frac { \pi }{ 2 } }{ sinx } dx=\)

(a)

1

(b)

2

(c)

-1

(d)

-2

\(\int _{ 0 }^{ 1 }{ \frac { 1 }{ 2x-3 } } \) dx = ____________

(a)

\(\frac { 1 }{ 2 } log3\)

(b)

\(-\frac { 1 }{ 2 } log3\)

(c)

\(\frac { 1 }{ 2 } log1\)

(d)

0

\(\int _{ 1 }^{ 4 }{ f\left( x \right) } dx,\) where f(x) =  \(\begin{cases} 7x+3\quad if\quad 1\le x\le 3 \\ 8x\quad if\quad 3\le x\le 4 \end{cases}\) is _____________.

(a)

58

(b)

60

(c)

62

(d)

52

∫ e^-t dt

(1)

\(\Gamma (n)\)

\(\int _{ 0 }^{ 1 }{ { e }^{ -t } } dt\quad \)

(2)

proper definite integer

\(\int _{ 0 }^{ \infty }{ { e }^{ -t } } dt\quad \)

(3)

\(\frac { n! }{ { a }^{ n+1 } } \)

For n > 0, \(\int _{ 0 }^{ \infty }{ { x }^{ n-1 }{ e }^{ -x } } \) dx

(4)

Improper definite intgral

\(\int _{ 0 }^{ \infty }{ { x }^{ n }{ e }^{ -ax } } \) dx where n is a positive integer

(5)

Indefinite inegral