The integral $\int e ^ { 2 x } \sin x d x$ is

A) $- \frac { e ^ { x } ( 2 \sin x - \cos x ) } { 5 } + C$
B) $\frac { e ^ { x } ( \sin x + 2 \cos x ) } { 5 } + C$
C) $\frac { e ^ { x } ( \sin x - 2 \cos x ) } { 5 } + C$
D) $\frac { e ^ { x } ( 2 \sin x + \cos x ) } { 5 } + C$
E) $\frac { e ^ { x } ( 2 \sin x - \cos x ) } { 5 } + C$

Question

The integral $\int e ^ { 2 x } \sin x d x$ is

A) $- \frac { e ^ { x } ( 2 \sin x - \cos x ) } { 5 } + C$
B) $\frac { e ^ { x } ( \sin x + 2 \cos x ) } { 5 } + C$
C) $\frac { e ^ { x } ( \sin x - 2 \cos x ) } { 5 } + C$
D) $\frac { e ^ { x } ( 2 \sin x + \cos x ) } { 5 } + C$
E) $\frac { e ^ { x } ( 2 \sin x - \cos x ) } { 5 } + C$

Quizplus · Accepted Answer

This integral can be solved using integration by parts or the tabular method, which involves integrating $e^{2x}$ and differentiating $\sin x$ repeatedly. The correct result, after simplifying, matches option E.

The Integral ∫e2xsin⁡xdx\int e ^ { 2 x } \sin x d x∫e2xsinxdx

The Integral $\int e ^ { 2 x } \sin x d x$