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

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Answer 1 · 2016-04-19T03:03:18

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

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

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

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

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