<p>If \(\int (\sin^3 \theta + \sin \theta) \cos \theta e^{\sin \theta} d\theta = (A \sin^3 \theta + B \cos^2 \theta + C \sin \theta + D \cos \theta + E) e^{\sin \theta} + F\), then:</p>
Step-by-Step Solution
Key Concept: Use substitution and integration by parts with exponential functions.
<p>Substituting $u = \sin \theta$, the integral becomes $\int (u^3 + u) e^u du$. Using integration by parts repeatedly and comparing coefficients with the given form yields specific values for $A, B, C, D, E$ that need verification.</p>
Correct Answer: d