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 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)

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

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

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