How do I find: $$\int \sin x \cos (\cos x) \cos(\sin x)+\cos x\sin(\cos x)\sin(\sin x)dx$$? I made this one up by taking the derivative of $-\sin (\cos x)\cos (\sin x)$, and I wonder how someone would solve this monster.
Find $\int \sin x \cos (\cos x) \cos(\sin x)+\cos x\sin(\cos x)\sin(\sin x)dx$
calculuscontest-mathindefinite-integralsintegrationtrigonometric-integrals
Best Answer
$$ \begin{aligned} \int \sin x \cos (\cos x) \cos (\sin x)= & -\int \cos (\sin x) d(\sin (\cos x))\\=& -\cos (\sin x) \sin (\cos x)-\int \sin (\cos x) \sin (\sin x) \cos x d x \quad \textrm{ (By IBP)} \end{aligned} $$ Removing the last integral to the left yields $$\int \left(\sin x \cos (\cos x) \cos (\sin x)+\cos x \sin (\cos x) \sin (\sin x) \right)dx= -\cos(\sin x) \sin (\cos x)+C $$