Expert: Paul Klarreich - 3/9/2007

Question
Hello, I need help finding the integral of
cos^2(2theta)  Ive been told that I can use a trig substitution

cos^(n(theta)) = 1 + cos2(n(theta)) / 2   is this correct or am I doing something wrong.  Thanks

Answer
Questioner:   Jeff
Category:  Calculus
 
Subject:  calculus 2
Question:  Hello, I need help finding the integral of
cos^2(2theta)  Ive been told that I can use a trig substitution

cos^(n(theta)) = 1 + cos2(n(theta)) / 2   is this correct or am I doing something wrong?  Thanks
..........................................
Hi, Jeff,

Yes, basically you are on the right track.  This is the 'half-angle' trick for integrating  cos^2(x) [and it works for sin^2(x), too]

The formulas:
          1 - cos 2A
sin^2(A) = -----------  
              2

          1 + cos 2A
cos^2(A) = -----------  
              2
are used to handle even powers of sine or cosine.

So your integral goes:(I'll use x instead of theta.)

{
| cos^2(2x) dx =
}

{ 1 + cos(4x)
| ----------- dx =
}     2

The rest should be routine.  The first term is just 1/2.  The second is a cos(u) substitution.  I'll leave that to you.  Any more trouble, send it along and I'll see what I can do.