Integrate cos^3x

To integrate cos^3x, also written as ∫cos³x dx, cos cubed x, cos^3(x), and (cos x)^3, we start by using standard trig identities to simplify the integral.

We factorise out one of the cosx terms to get a cosx squared term. As you can see, it means the same thing.

We recall the Pythagorean trig identity and rearrange it for the cosx squared term.

We can now eliminate the cos squared term by substitution of the rearranged trig identity.

Hence, we get this new expression.

Let u = sinx.

Then, du/dx = cosx.

We rearrange the previous expression for du. As you can see, cosx dx is exactly like the part outside the brackets in our integration problem. We can therefore replace it all with du.

On the LHS, we also  replace sin squared term inside the brackets with u to give a new expression on the RHS. It means the same thing but is in terms of u.

We rewrite our integration problem by integrating each term in the brackets separately. This means the same thing as our original integration problem.

We now integrate to give this intermediate result in terms of u. In the next step, we replace u with sinx.

Hence, this is the final answer.