TITLE: Show that $\int_0^\infty e^{-x}\cos x \text{dx}=\int_0^\infty e^{-x} \sin x \text{dx}$
QUESTION [5 upvotes]: Show that $\int_0^\infty e^{-x}\cos x \ \text{d}x=\int_0^\infty e^{-x} \sin x \ \text{d}x$ using integration by parts.

For the LHS, I got:
$-e^{-x} \cos x-\int e^{-x} \sin x \ \text{d}x$
I'm not sure how to show that the RHS integral is equal to this...

REPLY [13 votes]: $$\int_{0}^{+\infty}(\cos x-\sin x) e^{-x}\,dx = \left.e^{-x}\sin x\right|_{0}^{+\infty} = 0.$$