Evaluate the following integrals as the limits of sums.
\(\int _{ 0 }^{ 1 }{ (5x+4)dx } \)

Evaluate the following integrals as the limits of sums.
\(\int _{ 1 }^{ 2 }{( 4x^2-1)dx } \)

Evaluate the following definite integrals:
\(\int _{ 3 }^{ 4 }{ \frac { dx }{ { x }^{ 2 }-4 } } \)

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

Evaluate the following integrals using properties of integration:
\(\int _{ 0 }^{ 2\pi }{ xlog\left( \frac { 3+cos\ x }{ 3-cos\ x } \right) } dx\)

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

Evaluate the following integrals using properties of integration:
\(\int _{ 0 }^{ 1 }{ |5x-3|dx } \)

Evaluate the following integrals using properties of integration:
\(\int _{ 0 }^{ 1 }{ \frac { log(1+x) }{ 1+{ x }^{ 2 } } } dx\)

Evaluate the following
\(\int _{ 0 }^{ \pi /2 }{ { sin }^{ 10 }x\quad dx } \)

Evaluate the following
\(\int _{ 0 }^{ \pi /4 }{ { sin}^{ 6}2x\ dx } \)

Evaluate the following:
\(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \frac { { e }^{ -tanx } }{ { cos }^{ 6 }x } } dx\)

If \(\int _{ 0 }^{ \infty }{ { e }^{ -a{ x }^{ 2 } }{ x }^{ 3 }dx=32,\alpha >0,\ find\ \alpha } \)

Find the area of the region common to the circle x² +  y² = 16 and the parabola y² = 6x.

Find, by integration, the volume of the solid generated by revolving about the x-axis, the region enclosed by y = 2x², y = 0 and x = 1.

A watermelon has an ellipsoid shape which can be obtained by revolving an ellipse with major-axis 20 cm and minor-axis 10 cm about its major-axis. Find its volume using integration.