Step-by-Step Solution
Key Concept: General
$I = \int e^x \sin x dx = \sin x \cdot e^x - \int \cos x \cdot e^x dx = \sin x \cdot e^x - [\cos x \cdot e^x - \int (-\sin x) e^x dx] = \sin x \cdot e^x - \cos x \cdot e^x - I \Rightarrow 2I = \sin x \cdot e^x - \cos x \cdot e^x \Rightarrow I = \frac{1}{2} (\sin x - \cos x) e^x + c$
Correct Answer: $\frac{1}{2} (\sin x - \cos x) e^x + c$