Just Require Some Checking For My Differential

Question

If $f(x)=\cos(\sin(\cos(x)))$, $f'(x)=\sin(\sin(\cos(x)))\cos(\cos(x))\sin(x)$. Tried using Wolfram but it couldn't compute my integral. Am I correct?

Answer 1

Looks fine. To compute the original function from $f'(x)$ in this case you would proceed as follows:

$\int\sin(\sin(\cos(x)))\cos(\cos(x))\sin(x)dx$

Take $u=\sin(\cos(x))$ so that $du=-\cos(\cos(x))\sin(x)dx$ so that the integral becomes

$=-\int\sin(u)du=\cos(u)+C$. So the integral is $\cos(\sin(\cos(x)))+C$. We can evaluate the function at one point to remove the constant.