Integrating absolute value

Question

I have some trouble finding the right way to find integral of $$ \int | e^x \sin x| dx $$ How to do it?

Answer 1

Hint

You have to separate the cases where $x\in[2k\pi, \pi+2k\pi]$ and $x\in[-\pi+2k\pi, 2k\pi]$ and then you make a double part integration with $u(x)=e^x$ and $v'(x)=\sin x$.

In general, if you have to integrate on $A\subset \mathbb R$,

$$\int_A |e^x\sin x|dx=-\sum_{k}\int_{A\cap[-\pi+2k\pi,2k\pi]}e^x\sin xdx+\sum_{k}\int_{A\cap[2k\pi,\pi +2k\pi]}e^x\sin xdx,$$

and integrate by a double part integration each integral with $u(x)=e^x$ and $v'(x)=\sin x$.