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

1 Answer
Apr 19, 2016

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

Explanation:

We will want to make use of the following integral:

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

Thus, we want to set the interior of the sinsin function, which is in fact another sinsin function, equal to uu.

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

Thus, we have

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

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