Evaluate the integral.
-$$\int e ^ { 2 x } \cos 7 x d x$$
A) $\frac { e ^ { 2 x } } { 2 } [ \sin 7 x + \cos 7 x ] + C$
B) $\frac { e ^ { 2 x } } { 53 } [ 7 \sin 7 x - 2 \cos 7 x ] + C$
C) $\frac { e ^ { 2 x } } { 53 } [ 7 \sin 7 x + 2 \cos 7 x ] + C$
D) $\frac { 1 } { 53 } \left[ 7 e ^ { 2 x } \sin 7 x + 2 \cos 7 x \right] + C$

Question

Quizplus · Accepted Answer

This integral can be solved using integration by parts or the tabular method, which involves integrating $e^{2x}$ and differentiating $\cos 7x$ repeatedly. The correct answer involves a combination of $\sin 7x$ and $\cos 7x$ multiplied by $e^{2x}$, with coefficients that balance the derivatives of these functions. The coefficients $7$ and $2$ in option C correctly balance the integration by parts process, resulting in $\frac{e^{2x}}{53}[7\sin 7x + 2\cos 7x] + C$.

Evaluate the Integral ∫e2xcos⁡7xdx\int e ^ { 2 x } \cos 7 x d x∫e2xcos7xdx

Evaluate the Integral $\int e ^ { 2 x } \cos 7 x d x$