Evaluate the integral : `int_(0)^(pi/2) cos^5 x dx`

The value of
`int_(0)^(pi/2) cos ^5 x dx` has to be
determined.

`int_(0)^(pi/2) cos ^5 x
dx`

=> `int_(0)^(pi/2) cos^4
x*cos x dx`

=> class="AM">`int_(0)^(pi/2) (cos^2 x)^2 *cos x dx`

=> `int_(0)^(pi/2) (1 - sin^2x)^2 *cos x
dx`

=> `int_(0)^(pi/2) (1 -
2*sin^2x + sin^4x) *cos x dx`

Let class="AM">`sin x = y`

class="AM">`dy = cos x* dx`

class="AM">`sin 0 = 0` and `sin(pi/2)
= 1`

The required integral is changed to:

`int_(0)^(1) 1 - 2*y^2 + y^4
dy`

=> `y - 2y^3/3 +
y^5/5` between y = 0 and y = 1

=> class="AM">`1 - 2/3 + 1/5`

=> class="AM">`8/15`

The value of
`int_(0)^(pi/2) cos ^5 x dx` = class="AM">`8/15`