Integrate cos^23x

To integrate cos^23x, also written as ∫cos²3x dx, cos squared 3x, (cos3x)^2, and cos^2(3x), we start by using a u substitution.

Let u=3x.

Then du/dx=3.

We then rearrange the previous expression for dx.

We then rewrite the integration problem in terms of u by substituting out the dx, and 3x. On the RHS we have a simpler integration in terms of u, which means the same thing.

We simplify by moving the constant 1/3 outside of the integral. As you can see, we now need to integrate sin²u.

As it just so happens, I have already shown how to integrate cos^2x, and therefore we can use the above result that we previously obtained.

Therefore in our case, it is in terms of u as shown on the RHS.

We then simplify and substitute back the 2x for u.

We then simplify further, and this is the answer.