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Integration Hints

Date: 02/23/99 at 19:08:03
From: Scott
Subject: Question on Integration

Can you help me integrate:


I have spent a full hour and I keep getting stuck, at a different point 
each time, with a different answer. I think you have to split it up and 
use identities instead, but I do not know which ones to use.

If you could help me out with this, by writing me either the work, or 
a step-by-step proccess of how to solve this, I would GREATLY 
appreciate it.  

Date: 02/23/99 at 19:37:16
From: Doctor Wilkinson
Subject: Re: Question on Integration

I am not going to do all the work, but I can give you some useful 
hints. There are a lot of ways to go about this, but one of the 
simplest is to start by using the identity sin^2 x + cos^2 x = 1, and 
rewriting the integral as the integral of

  cos^2 x cos^2 x dx = cos^2 x (1 - sin^2 x) dx, 

which is the integral of

  cos^2 x dx - the integral of sin^2 x cos^2 x dx.

Next I would use the double-angle formulae

  cos(2x ) = cos^2 x - sin^2 x = 2 cos^2 x - 1

to express the first integral in terms of the integral of cos (2x) dx


  sin(2x) = 2 sinx cos x

to express the second integral in terms of the integral of sin^2(2x), 
which again can be expressed in terms of the integral of cos(2x) dx by 
using the same double-angle formula for the cosine in the form

  cos(2x) = 1 - 2 sin^2 x.

I hope this helps. You have to proceed carefully and watch the signs 
and so on.

- Doctor Wilkinson, The Math Forum   
Associated Topics:
High School Calculus
High School Trigonometry

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