The value of \(\int _{ -4 }^{ 4 }{ \left[ { tan }^{ -1 }\left( \frac { { x }^{ 2 } }{ { x }^{ 4 }+1 } \right) +{ tan }^{ -1 }\left( \frac { { x }^{ 4 }+1 }{ { x }^{ 2 } } \right) \right] dx } \) is

(a)

\(\pi\)

(b)

\(2\pi\)

(c)

\(3\pi\)

(d)

\(4\pi\)

The value of \(\int _{ -\frac { \pi }{ 4 } }^{ \frac { \pi }{ 4 } }{ \left( \frac { { 2x }^{ 7 }-{ 3x }^{ 5 }+{ 7x }^{ 3 }-x+1 }{ { cos }^{ 2 }x } \right) dx } \) is 

(a)

4

(b)

3

(c)

2

(d)

0

If \(f(x)=\int_{0}^{x} t \cos t d t, \text { then } \frac{d f}{d x}=\)

(a)

cos x - x sin x

(b)

sin x + x cos x

(c)

x cos x

(d)

x sin x

The area between y² = 4x and its latus rectum is

(a)

\(\frac{2}{3}\)

(b)

\(\frac{4}{3}\)

(c)

\(\frac{8}{3}\)

(d)

\(\frac{5}{3}\)

The value of \(\int _{ 0 }^{ 1 }{ x{ (1-x) }^{ 99 }dx } \) is

(a)

\(\frac{1}{11000}\)

(b)

\(\frac{1}{10100}\)

(c)

\(\frac{1}{10010}\)

(d)

\(\frac{1}{10001}\)

The value of \(\int _{ 0 }^{ \pi }{ \frac { dx }{ 1+{ 5 }^{ cos\ x } } } \) is

(a)

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

(b)

\(\pi\)

(c)

\(\frac{3\pi}{2}\)

(d)

\(2\pi\)

If \(\frac{\Gamma(n+2)}{\Gamma(n)}=90\) then n is 

(a)

10

(b)

5

(c)

8

(d)

9

The value of \(\int _{ 0 }^{ \frac { \pi }{ 6 } }{ { cos }^{ 3 }3x\ dx }\ is\)

(a)

\(\frac{2}{3}\)

(b)

\(\frac{2}{9}\)

(c)

\(\frac{1}{9}\)

(d)

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

The value of  \(\int _{ 0 }^{ \pi }{ { sin }^{ 4 }xdx } \) is

(a)

\(\frac{3\pi}{10}\)

(b)

\(\frac{3\pi}{8}\)

(c)

\(\frac{3\pi}{4}\)

(d)

\(\frac{3\pi}{2}\)

The value of \(\int _{ 0 }^{ \infty }{ { e }^{ -3x }{ x }^{ 2 }dx } \) is

(a)

\(\frac{7}{27}\)

(b)

\(\frac{5}{27}\)

(c)

\(\frac{4}{27}\)

(d)

\(\frac{2}{27}\)

If \(\int _{ 0 }^{ a }{ \frac { 1 }{ 4+{ x }^{ 2 } } dx=\frac { \pi }{ 8 } } \) then a is

(a)

4

(b)

1

(c)

3

(d)

2

The volume of solid of revolution of the region bounded by y² = x(a − x) about x-axis is

(a)

\(\pi a^3\)

(b)

\(\frac { \pi { a }^{ 3 } }{ 4 } \)

(c)

\(\frac { \pi { a }^{ 3 } }{ 5 } \)

(d)

\(\frac { \pi { a }^{ 3 } }{ 6 } \)

If \(f(x)=\int_{1}^{x} \frac{e^{\sin u}}{u} d u, x>1 \text { and }\int_{1}^{3} \frac{e^{\sin x^{2}}}{x} d x=\frac{1}{2}[f(a)-f(1)]\), then one of the possible value of a is

(a)

3

(b)

6

(c)

9

(d)

5

The value of \(\int _{ 0 }^{ 1 }{ { ({ sin }^{ -1 }x) }^{ 2 } } dx\) is

(a)

\(\frac { { \pi }^{ 2 } }{ 4 } -1\)

(b)

\(\frac { { \pi }^{ 2 } }{ 4 } +2\)

(c)

\(\frac { { \pi }^{ 2 } }{ 4 } +1\)

(d)

\(\frac { { \pi }^{ 2 } }{ 4 } -2\)

The value of \(\int _{ 0 }^{ a }{ { (\sqrt { { a }^{ 2 }-{ x }^{ 2 } } ) }^{ 3 } } dx\) is

(a)

\(\frac { { \pi a }^{3 } }{ 16 } \)

(b)

\(\frac { 3\pi { a }^{ 4 } }{ 16 } \)

(c)

\(\frac { 3\pi { a }^{2 } }{ 8} \)

(d)

\(\frac { 3\pi { a }^{ 4 } }{ 8} \)

If \(\int _{ 0 }^{ x }{ f(t)dt=x+\int _{ x }^{ 1 }{ tf } (t)dt } \), then the value of f (1) is

(a)

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

(b)

2

(c)

1

(d)

\(\frac{3}{4}\)

\(\text { The value of } \int_{0}^{\frac{2}{3}} \frac{d x}{\sqrt{4-9 x^{2}}} \text { is }\)

(a)

\(\frac{\pi}{6}\)

(b)

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

(c)

\(\frac{\pi}{4}\)

(d)

\({\pi}\)

The value of \(\int _{ -1 }^{ 2 }{ |x|dx } \) is

(a)

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

(b)

\(\frac{3}{2}\)

(c)

\(\frac{5}{2}\)

(d)

\(\frac{7}{2}\)

For any value of \(n \in \mathbb{Z}, \int_{0}^{\pi} e^{\cos ^{2} x} \cos ^{3}[(2 n+1) x] d x\) is

(a)

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

(b)

\(\pi\)

(c)

0

(d)

2

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

(a)

\(\frac{3}{2}\)

(b)

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

(c)

0

(d)

\(\frac{2}{3}\)