How do you find the antiderivative of #cos^2 (x)#?

1 Answer
Jul 31, 2016

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

Explanation:

The trick to finding this integral is using an identity--here, specifically, the cosine double-angle identity.

Since #cos(2x)=cos^2(x)-sin^2(x)#, we can rewrite this using the Pythagorean Identity to say that #cos(2x)=2cos^2(x)-1#. Solving this for #cos^2(x)# shows us that #cos^2(x)=(cos(2x)+1)/2#.

Thus:

#intcos^2(x)dx=1/2intcos(2x)+1dx#

We can now split this up and find the antiderivative.

#=1/2intcos(2x)dx+1/2int1dx#

#=1/4int2cos(2x)dx+1/2x#

#=1/4sin(2x)+1/2x+C#