Question

What is the antiderivative of `cos(x)sin(sin(x))dx`?

Answer 1

`-cos(sin(x))+C`

Explanation:

We will want to make use of the following integral:

`intsin(u)du=-cos(u)+C`

Thus, we want to set the interior of the `sin` function, which is in fact another `sin` function, equal to `u`.

So, we want to set `u=sin(x)`. This implies that `(du)/dx=cos(x)` and `du=cos(x)dx`.

Thus, we have

`intcos(x)sin(sin(x))dx=intsin(sin(x))*cos(x)dx`

`=intsin(u)du=-cos(u)+C=-cos(sin(x))+C`