$\int e^x\sin x\ dx$: how many substitutions?

Question

I got the answer of $-\frac12e^x\cos x$ after substituting twice. Do I always have to substitute twice for all simple integration-by-parts problems?

Answer 1

Hint:

$$\int e^x \sin{x} dx=\frac{1}{2} \int e^x((\sin{x}-\cos{x})+(\cos{x}+\sin{x}))dx$$

Then, use $$\int e^x(f(x)+f'(x))dx=e^xf(x)+c$$