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`

=> `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 `sin x = y`

`dy = cos x* dx`

`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

=> `1 - 2/3 + 1/5`

=> `8/15`

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