Evaluate the integral.
-$$\int \mathrm { e } ^ { 3 x } \cos 2 \mathrm { xdx }$$
A) $\frac { 1 } { 13 } \left[ 2 \mathrm { e } ^ { 3 \mathrm { x } } \sin 2 \mathrm { x } + 3 \cos 2 \mathrm { x } \right] + \mathrm { C }$
B) $\frac { e ^ { 3 x } } { 13 } [ 2 \sin 2 x - 3 \cos 2 x ] + C$
C) $\frac { e ^ { 3 x } } { 13 } [ 2 \sin 2 x + 3 \cos 2 x ] + C$
D) $\frac { \mathrm { e } ^ { 3 x } } { 2 } [ \sin 2 x + \cos 2 x ] + C$

Question

Quizplus · Accepted Answer

This integral can be solved using integration by parts or the tabular method, which involves integrating $e^{3x}$ and differentiating $\cos 2x$ repeatedly. The correct solution involves a combination of $e^{3x}$ multiplied by a linear combination of $\sin 2x$ and $\cos 2x$, with coefficients that ensure the derivative of the answer matches the integrand. The coefficients $2$ and $3$ correctly balance the effects of differentiating $e^{3x}$ and the trigonometric functions, and dividing by $13$ comes from the constants involved in integration by parts, specifically from the sum of squares of the coefficients in the exponential and trigonometric functions (i.e., $3^2 + 2^2 = 13$).

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Evaluate the Integral $\int \mathrm { e } ^ { 3 x } \cos 2 \mathrm { xdx }$