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

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

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

If \(\int _{ 0 }^{ 2a }{ f(x) } dx=2\int _{ 0 }^{ a }{ f(x) } \) then __________

The ratio of the volumes generated by revolving the ellipse \(\frac { { x }^{ 2 } }{ 9 } +\frac { { y }^{ 2 } }{ 4 } \) = 1 about major and minor axes is __________

Evaluate the following definite integrals:
\(\int _{ -1 }^{ 1 }{ \frac { dx }{ { x }^{ 2 }+2x+5 } } \)

Evaluate the following integrals using properties of integration:
\(\int _{ 0 }^{ 2\pi }{ { sin }^{ 4 }{ x\ cos }^{ 3 }xdx } \)

Evaluate the following:
\(\int _{ 0 }^{ 1 }{ { x }^{ 3 }{ e }^{ -2x }dx } \)

Evaluate \(\int _{ 0 }^{ 1 }{ \frac { { e }^{ x } }{ 1+{ e }^{ 2x } } dx } \)

Find the area of the region enclosed by the curve y = \(\sqrt x\) + 1, the axis of x and the lines x = 0, x = 4.

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

Evaluate:  \(\int ^{\frac{\pi}{2}}_{\frac{\pi}{2}}\)x cos x dx.

Find, by integration, the volume of the solid generated by revolving about y-axis the region bounded between the parabola x = y² +1, the y-axis, and the lines y = 1 and y = −1.

Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { sin \ x }{ 9+{ cos }^{ 2 } } dx } \)

Evaluate \(\int _{ 0 }^{ 1 }{ \sqrt { 9-4{ x }^{ 2 } } dx } \)

Evaluate \(\int ^\frac {\pi}{2}_{0} \)( sin² x + cos⁴ x ) dx

Find the area of the region bounded by x−axis, the curve y = |cos x|, the lines x = 0 and x = \(\pi\).