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

\(\int \frac{\sin 5 x-\sin x}{\cos 3 x} d x\) is _______.

(a)

−cos 2x + c

(b)

−cos 2x + c

(c)

\(-\frac14\)cos2x + c

(d)

−4cos2x + c

ഽ\(\frac{logx}{x}\) dx , x > 0 is _______.

(a)

\(\frac12\)(log x)² + c

(b)

-\(\frac12\)(log x)²

(c)

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

(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\(\sqrt { { e }^{ x } } \) dx is _______.

(a)

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

(b)

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

(c)

\(\frac12\sqrt { { e }^{ x } } +c\)

(d)

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

ഽe^2x[2x² + 2x]dx _______.

(a)

e^2xx² + c

(b)

xe^2x + c

(c)

2x²e² + c

(d)

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

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

ഽ\(\left[ \frac { 9 }{ x-3 } -\frac { 1 }{ x+1 } \right] \)dx is _______.

(a)

\(log\left| x-3 \right|-log \left| x+1 \right| +c\)

(b)

\(log\left| x-3 \right|+log \left| x+1 \right| +c\)

(c)

\(9log\left| x-3 \right|-log \left| x+1 \right| +c\)

(d)

\(9log\left| x-3 \right|+log \left| x+1 \right| +c\)

ഽ\(\frac { { 2x }^{ 3 } }{ 4+{ x }^{ 4 } } \)dx is _______.

(a)

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

(b)

\(\frac { 1 }{ 2 } log\left| 4+{ x }^{ 4 } \right| +c\)

(c)

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

(d)

\(log\left| \frac { { 2x }^{ 3 } }{ { 4+x }^{ 4 } } \right| +c\)

ഽ\(\frac { dx }{ \sqrt { { x }^{ 2 }-{ 36 } } } \) is _______.

(a)

\(\sqrt { { x }^{ 2 }-{ 36 } } +c\)

(b)

log\(\left| x+\sqrt { { x }^{ 2 }-36 } \right| +c\)

(c)

log\(\left| x-\sqrt { { x }^{ 2 }-36 } \right| +c\)

(d)

\(log\left| { x }^{ 2 }+\sqrt { { x }^{ 2 }-36 } \right| +c\)

ഽ\(\frac { 2x+3 }{ \sqrt { x^{ 2 }+3x+2 } } \) dx is _______.

(a)

\(\sqrt { x^{ 2 }+3x+2 } \) + c

(b)

\(2​​\sqrt { x^{ 2 }+3x+2 } +c\)

(c)

log(x² + 3x + 2)+ c

(d)

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

\(\int _{ 0 }^{ 1 }{ (2x+1) } dx\) is _______.

(a)

1

(b)

2

(c)

3

(d)

4

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

(a)

log 4

(b)

0

(c)

log 2

(d)

log 8

\(\int _{ 0 }^{ \infty }{ { e }^{ -2x } } \) dx is _______.

(a)

0

(b)

1

(c)

2

(d)

\(\frac12\)

\(\int _{ -1 }^{ 1 }{ { x }^{ 3 }{ e }^{ { x }^{ 4 } } } \) dx is _______.

(a)

1

(b)

2\(\int _{ -1 }^{ 1 }{ { x }^{ 3 }{ e }^{ { x }^{ 4 } } } \)dx

(c)

0

(d)

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

If f (x) is a continuous function and a < c < b, then \(\int_{a}^{c} f(x) d x+\int_{c}^{b} f(x) d x\) is _______.

(a)

\(\int_{a}^{b} f(x) d x-\int_{a}^{c} f(x) d x\)

(b)

\(\int_{a}^{c} f(x) d x-\int_{a}^{b} f(x) d x\)

(c)

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

(d)

0

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

(a)

0

(b)

2

(c)

1

(d)

4

\(\int _{ 0 }^{ 1 }{ \sqrt { { x }^{ 4 }({ 1-x) }^{ 2 } } } dx\) is _______.

(a)

\(\frac{1}{12}\)

(b)

\(\frac{-7}{12}\)

(c)

\(\frac{7}{12}\)

(d)

\(\frac{-1}{12}\)

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

The value of \(\int _{ 2 }^{ 3 }{ f(5-x) } dx-\int _{ 2 }^{ 3 }{ f(x) } dx\) is _______.

(a)

1

(b)

0

(c)

-1

(d)

5

\(\int _{ 0 }^{ 4 }{ \left( \sqrt { x } +\frac { 1 }{ \sqrt { x } } \right) } \)dx is _______.

(a)

\(\frac{20}{3}\)

(b)

\(\frac{21}{3}\)

(c)

\(\frac{28}{3}\)

(d)

\(\frac{1}{3}\)

\(\int _{ 0 }^{ \frac { \pi }{ 3 } }\)tanx dx is _______.

(a)

log 2

(b)

0

(c)

log\(\sqrt { 2 } \)

(d)

2 log 2

Using the factorial representation of the gamma function, which of the following is the solution for the gamma function \(\Gamma \)(n) when n = 8 _______.

(a)

5040

(b)

5400

(c)

4500

(d)

5540

\(\Gamma (n)\) is _______.

(a)

(n −1)!

(b)

n!

(c)

\(n\Gamma (n)\)

(d)

(n −1)\(\Gamma \)(n)

\(\Gamma (1)\) is _______.

(a)

0

(b)

1

(c)

n

(d)

n!

If n > 0, then \(\Gamma \)(n) is _______.

(a)

\(\int _{ 0 }^{ 1 }{ { e }^{ -x }x^{ n-1 } } \) dx

(b)

\(\int _{ 0 }^{ 1 }{ { e }^{ -x }{ x }^{ n } } dx\)

(c)

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

(d)

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

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

(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