find the indefinite integral using a trigonometric identity to set up a u-substitution

Question

∫sin⁴xcos⁵x dx

Answer 1

Start by using the Pythagorean identity to replace cos⁴ with (1- sin²)² = (1-2 sin² + sin⁴) .

The integrand is now

 

sin⁴(1- 2 sin² + sin⁴) cos = ( sin⁴ - 2 sin⁶ + sin⁸) cos

 

This can be integrated term by term using the u substitution u = sin.

However, it is easier to notice the cos factor is an integrating factor for each term.

The antiderivative can be written down right away.  It is

 

F  =  (1/5) sin⁵ - (2/7) sin⁷ + (1/9) sin⁹ + k