Integrate 12sin^2tcost

To integrate 12sin^2tcost, also written as ∫12sin²t cost dt, 12sin^2(t)cos(t), we start by rearranging the expression.

We simply move the constant outside of the integral sign to simplify the expression.

Let u = sint.

Then, du/dt = cost.

We rearrange the expression for du.

We replace cost dt with du, and sint with u to give a new expression on the RHS.

We then integrate u squared as normal, and introduce C as the constant.

We simplify and replace u with sint.

Hence, this is the answer.