Integration of Trigonometric Functions

Date: 05/28/99 at 09:14:00
From: Corenne
Subject: Integration in trigonometry

I am having problems with the following question:

Integrate sin(x)sin(2x).dx

I have tried integration by parts but it has not seemed to work.

   u  = sin(x)
   u' = -cos(x)
   v' = sin(2x)
   v  = 0.5cos(2x)

   0.5sin(x)cos(2x)-(integral of)[-0.5cos(x)cos(2x).dx]

Please help!
Thank you,

Corenne

Date: 05/28/99 at 10:17:50
From: Doctor Wilkinson
Subject: Re: Integration in trigonometry

Hi, Corenne. Integration by parts is a useful tool, but it's not a 
magic wand. A good question to ask is "what's the main problem here"?  

I'd say the main problem is that you have both sin(x) and sin(2x) and 
they don't mix well. Fortunately, we have a formula for sin(2x) in 
terms of sin(x) and cos(x). When you do integration involving trig 
functions, you often need to apply trigonometry and not just calculus.  
Writing

   sin(2x) = 2sin(x)cos(x)   gives you the integral of

   2sin^2(x) cos(x) dx

and now things are looking good, since cos(x) is the derivative of 
sin(x).

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