Question

What is the integral of `sin(x) cos(x)`?

Answer 1

It's `1/2 sin^2 (x) + C`.

The substitution used to solve this integral is simple.

Note that `cos(x)` is the derivative of `sin(x)`.

Define the variable `u=sin(x)`.
We have `du=cos(x)dx`.
So, `dx=1/cos(x) du`.

The integral:

`int sin(x) cos(x) dx = int u cos(x)/cos(x) du = int u du`

Knowing that `d/(du)[1/2 u^2 + C]=u` we have:

`int u du = int d/(du)[1/2 u^2 + C] du`

Using the Fundamental Theorem of Calculus we get:

`int d/(du)[1/2 u^2 + C] du = 1/2 u^2 + C = 1/2 sin^2 (x) + C`