Integrate 16cos^2x

To integrate 16cos^2x, also written as ∫16cos²x dx, 16cos squared x, 16cos^2(x), we start by using standard trig identities to to change the form.

We recall the Pythagorean trig identity and rearrange it for cos squared x.

We recall the double angle trig identity and rearrange it for sin squared x.

We substitute trig identity [2] into trig identity [1]. The substituted part is in brackets. We then simplify, and transpose for cos squared x.

Going back to our original integration problem, we can now rewrite it in a new form as shown on the LHS.

We can now multiply each term by 16 to simplify further.

We can now integrate each term separately.

We move the constant 8 outside the integral to simplify further. Then we focus on integrating the cos2x term as shown below.

Shown above is the answer for integrating cos2x. We can now substitute that part in our original integration problem.

This was the new form of our integration problem and we have the answer for the second term. The first term which is a constant 8 is simple to integrate.

Hence, we integrate the first term and substitute the solution for the second term.

Hence, this is the final answer.