Learning Trigonometric Substitution

[1MileCrash]

Homework Statement

My book has given me this example, with a step by step, but one of the steps have left me dumbfounded, they seem to have left a huge gap in steps..

Homework Equations

The Attempt at a Solution

$\int sin^{5}xdx$

$\int sin^{4}x sinx dx$

$\int (1-cos^{2}x)^{2} sinx dx$

$\int (1 - 2cos^{2} x + cos^{4} x) sinx dx$

Cool story bro. Here's the next step, what happened?

$= - cos x + \frac{2}{3}cos^{3}x - \frac{1}{5}cos^{5} x + C$

Did they bring a negative out of the integral and do some kind of U substitution?

 

[LCKurtz]

They put u = cos(x), du = -sin(x)dx. Try it.