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)

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

ഽ2^xdx is _______.

(a)

2^x log 2 + c

(b)

2^x + c

(c)

\(\frac { 2^{ x } }{ log2 } +c\)

(d)

\(\frac { log2 }{ { 2 }^{ x } } +c\)

ഽ\(\frac { sin2x }{ 2sinx } dx\) is _______.

(a)

sin x + c

(b)

\(\frac12\)sin x + c

(c)

cos x + c

(d)

\(\frac12\)cos x + c

\(\Gamma \left( \frac { 3 }{ 2 } \right) \) _______.

(a)

\(\sqrt { \pi } \)

(b)

\(\frac { \sqrt { \pi } }{ 2 } \)

(c)

\(2\sqrt { \pi } \)

(d)

\(\frac32\)

\(\int _{ 0 }^{ \infty }{ { x }^{ 4 }{ e }^{ -x } } \)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)

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

\(\int { { e }^{ x } } \) (1-cot x +cot² x) dx = _______________ +c

(a)

e^x cot x

(b)

- e^x cot x

(c)

e^x cosec x

(d)

-e^x cosec x

∫ e^-t dt

(1)

Improper definite intgral

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

(2)

proper definite integer

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

(3)

Indefinite inegral

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

(4)

\(\Gamma (n)\)

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

(5)

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

Integrate the following with respect to x.
\(\sqrt { 3x+5 } \)

 Integrate the following with respect to x.
\({ \left( { 9x }^{ 2 }-\frac { 4 }{ { x }^{ 2 } } \right) }^{ 2 }\)

 Integrate the following with respect to x.
\(\sqrt{x}\)(x³ − 2x + 3)

Evaluate \(\int { { a }^{ 3{ log }_{ a }x } } dx\)

Evaluate \(\int { \frac { 2{ cos }^{ 2 }x-cos2x }{ { sin }^{ 2 }x } } \)

 Evaluate \(\int \frac{a x^{2}+b x+c}{\sqrt{x}} d x\)

Evaluate \(\int \sqrt{2 x+1} \ d x\)

Evaluate \(\int { \frac { dx }{ { \left( 2x+3 \right) }^{ 2 } } } \)

Evaluate  \(\int { \frac { { { x }^{ 4 }+{ x }^{ 4 }+1 } }{ { x }^{ 2 }-x+1 } } \)

Evaluate \(\int { \frac { cos2x-cos2\alpha }{ cosx-cos\alpha } } dx\)

Evaluate \(\int { \frac { { 3x }^{ 2 }+2x+1 }{ x } dx } \)

If f'(x) = a sin x + b cos x and f'(0) = 4, f(0) = 3, f\(\left( \frac { \pi }{ 2 } \right) \) = 5, find f(x).