Evaluate \(\int _{ 0 }^{ x }{ { x }^{ 2 } } \)cos nx dx, where n is a positive integer.

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

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

Evaluate the following
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { sin }^{ 2 }x{ cos }^{ 4 }xdx } \)

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

Find, by integration, the volume of the solid generated by revolving about the y-axis, the region enclosed by x² = 1+ y and y = 3.

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

Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { e }^{ 2x }cosxdx } \)

Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { e }^{ -x } } \)

Evaluate \(\int _{ 0 }^{ 1 }{ \left( \frac { { e }^{ 5logx }-{ e }^{ 4logx } }{ { e }^{ 3logx }-{ e }^{ 2logx } } \right) } \)

If \(\int _{ 0 }^{ \infty }{ \frac { { x }^{ 2 }dx }{ \left( { x }^{ 2 }+{ a }^{ 2 } \right) \left( { x }^{ 2 }+{ b }^{ 2 } \right) \left( { x }^{ 2 }+{ c }^{ 2 } \right) } } =\frac { \pi }{ 2(a+b)(b+c)(c+a) } \) then find \(\int _{ 0 }^{ \infty }{ \frac { dx }{ \left( { x }^{ 2 }+4 \right) \left( { x }^{ 2 }+9 \right) } } \)

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

Find the area bounded by y=x²+2,x-x-axis, x=1 and x=2

Find the area enclosed between the parabola y2=4ax and the line x=a,x=9a.

Find the area bounded by the curve y=cosax in one arc of the curve.