Expert: Paul Klarreich - 5/28/2008

Question
I'm stuck on this one integral of {cos^4xdx, the exponent is even, so do I use double-angle identities? I tried it, but it makes it complicated.

Answer
Questioner:   Lana
Category:  Calculus
Private:  No
 
Subject:  trig identity integration
Question:  I'm stuck on this one integral of {cos^4xdx, the exponent is even, so do I use double-angle identities? I tried it, but it makes it complicated.
..................................
Hi, Lana,

If you are referring to :
          1 + cos (2t)
cos^2(t) = ------------
              2
then yes, you use it -- twice. [Actually this is usually called a 'half-angle' identity, but who's counting?]

Yes, it does get complicated, but you are a big girl now (I assume) and I'm sure you'll be OK.

cos^4(x) = (cos^2(x))^2

            1 + cos (2x)
cos^4(x) = (---------------)^2     << first use;  t = x
                 2

           1 + 2 cos (2x) + cos^2(2x)
cos^4(x) = ---------------------------   << square it out.
                    4

           1 + 2 cos (2x) + (1 + cos(4x))/2
cos^4(x) = ---------------------------------   << second use;  t = 2x
                    4
=  1/4(1 + 2 cos (2x) + 1/2(1 + cos(4x))/2), which integrates to:

1/4(x + sin (2x) + 1/2(x + sin(4x)/4)

Now just put it all together.

P.S. Check THE INTEGRATOR web site:

integrals.wolfram.com/index.jsp