Evaluate the improper integral $\int_{-\infty}^0 {e^x}\text{sin}\, xdx$

Question

Evaluate the improper integral $$\int_{-\infty}^0 {e^x} \sin x \, dx$$ justifying any non-trivial steps.

I have begun to answer this question by finding the indefinite integral, $$\int {e^x} \sin x \, dx = \dfrac{1}2{e^x}(\sin x-\cos x)+C.$$

then gone on to say,

$$\lim_{a\rightarrow {-\infty}} \int_a^0 e^x \sin x\,dx.$$

Not too sure where to go from here.

Answer 1

You have

$$\lim_{a\to -\infty} \int_a^0 e^x \sin x \; dx = \left.\lim_{a\to -\infty}\frac{e^x(\sin x - \cos x)}{2}\right|_a^0 $$ $$=\lim_{a\to -\infty}\frac{e^0(\sin 0 - \cos 0)}{2}-\frac{e^a(\sin a - \cos a)}{2} =-\frac{1}{2}-0.$$