Question

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

Answer 1

`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`