Find the integral.
$$\int e ^ { 3 x } \sin 5 x d x$$
Select the correct answer.
A) $\frac { 1 } { 34 } e ^ { 3 x } ( 3 \sin 5 x - 3 \cos 5 x ) - C$

B) $\frac { 1 } { 34 } e ^ { 3 x } ( 3 \sin 5 x - 5 \cos 3 x ) - C$

C) $\frac { 1 } { 34 } e ^ { 3 x } ( 3 \sin 5 x - 3 \cos 5 x ) + C$

D) $\frac { 1 } { 34 } e ^ { 3 x } ( 3 \sin 5 x - 5 \cos 5 x ) + C$

Question

Find the integral.
$$\int e ^ { 3 x } \sin 5 x d x$$
Select the correct answer.
A) $\frac { 1 } { 34 } e ^ { 3 x } ( 3 \sin 5 x - 3 \cos 5 x ) - C$

B) $\frac { 1 } { 34 } e ^ { 3 x } ( 3 \sin 5 x - 5 \cos 3 x ) - C$

C) $\frac { 1 } { 34 } e ^ { 3 x } ( 3 \sin 5 x - 3 \cos 5 x ) + C$

D) $\frac { 1 } { 34 } e ^ { 3 x } ( 3 \sin 5 x - 5 \cos 5 x ) + C$

Quizplus · Accepted Answer

We need to use integration by parts. Let
$$u = e^{3x} \quad \Rightarrow \quad u' = 3e^{3x}$$
$$dv = \sin 5x dx \quad \Rightarrow \quad v = -\frac{1}{5} \cos 5x$$
Then we have
\begin{align*}
\int e^{3x} \sin 5x dx &= -\frac{1}{5} e^{3x} \cos 5x + \frac{3}{5} \int e^{3x} \cos 5x dx \
&= -\frac{1}{5} e^{3x} \cos 5x + \frac{3}{5} \left( \frac{1}{3}e^{3x} \sin 5x - \frac{1}{3} \int e^{3x} \sin 5x dx ight) \
&= -\frac{1}{5} e^{3x} \cos 5x + \frac{1}{5} e^{3x} \sin 5x - \frac{1}{5} \int e^{3x} \sin 5x dx \
\end{align*}
Solving for $\int e^{3x} \sin 5x dx$ gives
$$\int e^{3x} \sin 5x dx = \frac{1}{34} e^{3x} (3 \sin 5x - 5 \cos 5x) + C$$
Therefore, the answer is $\boxed{	extbf{(D)}}$.

Find the Integral ∫e3xsin⁡5xdx\int e ^ { 3 x } \sin 5 x d x∫e3xsin5xdx

Find the Integral $\int e ^ { 3 x } \sin 5 x d x$