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