A Change of Variables

Date: 05/05/2003 at 01:12:07
From: Desire
Subject: Trigonometric identities

How do I solve int(sin^5(2x)cos(2x)dx)?

Date: 05/05/2003 at 03:40:57
From: Doctor Luis
Subject: Re: Trigonometric identities

Hello,

You are solving the following integral:

   /
  |
  | (sin(2x))^5 * cos(2x) dx
  |
 /

One important thing to notice right away is that the derivative of the 
sin(2x) is cos(2x), and that is the key to solving this problem (a 
simple change of variables).

As you might have guessed, the simplest substitution you can make is 
u = sin(2x), which will give you du = 2*cos(2x)dx

After substituting your change of variable from x to u, the integral 
looks like this:

   /
  |
  | u^5 * (du/2)
  |
 /

Note that I substituted u for sin(x), and du/2 for cos(2x)dx.

This should be a form that you can integrate more easily.

Let us know if you have more questions.

- Doctor Luis, The Math Forum
  http://mathforum.org/dr.math/