Integrate sin^3x

To integrate sin^3x, also written as ∫sin³x dx, sin cubed x, sin^3(x), and (sin x)^3, we start by using standard trig identities to simplify the integral.

If we take one of the factors out, then we can express the integration in a different form, yet it means the same thing.

We then recall this Pythagorean trig identity.

We rearrange it for sin²x.

We substitute the rearranged trig identity for sin²x in our integration problem to give a new expression that means the same thing.

Let u = cosx.

Then du/dx = -sinx

We rearrange it to give an expression for du.

As you can see, our expression for du is almost the same as that in our integration problem, except du has a negative sign. We can solve this issue with a mathematical trick as shown below.

If we were to multiply -1 with -1 then the result is +1 and we have not changed anything. That is exactly what we are doing here. We place a -1 outside the integral and another inside the integral. As you can see, we now have -sinx dx, and therefore we can do a straight swap with du.

We can now rewrite the whole integration in terms of u, by replacing cosx with u.

The -1 that was behind the integral can be multiplied out, thereby reversing the signs on the LHS. This is the same as the expression on the RHS where we integrate each term separately w.r.t. du.

Hence our integration solution is in terms of u is as shown above, and C is the final integration constant.

We replace u with cosx to give the final solution.