# Math Help - [SOLVED] Integration by Substitution

1. ## [SOLVED] Integration by Substitution

I am trying to integrate the equation sin (x) * cos(cos (x)), I assume it is best done through integration by U-substitution but I cannot seem to come up with the U. Any help would be much appreciated, thanks!

2. Hello, MikeDube!

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

We have: . $\int\cos(\cos x)\cdot(\sin x\,dx)$

Let: $u \,=\,\cos x \quad\Rightarrow\quad du \,=\,-\sin x\,dx \quad\Rightarrow\quad \sin x\,dx \,=\,-du$

Substitute: . $\int \underbrace{\cos(\cos x)}_{\downarrow}\cdot\underbrace{(\sin x\,dx)}_{\downarrow}$
. . . . . . . . . . $\int\; \cos u\qquad (-du)\quad=\quad - \int\cos u\,du$

Got it?