Step-by-Step Solution
Key Concept: Use substitution with $e^x\sin x$ as the intermediate variable to convert the integral into the standard arctangent form.
Let $I = \int \frac{e^x\cos x}{1 + e^{2x}\sin^2 x}dx$. Substituting $e^x t$ where $e^x\sin x = t$, we get $\frac{e^x dx}{e^x dt}$. Thus $I = 2\int \frac{dt}{1+t^2} = 2\tan^{-1}(t) + C = 2\tan^{-1}(e^x\sin x) + C$.
Correct Answer: C