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Functions

∫ e2xsin 3x d...

Question

$\int e^{2 x} sin 3 x d x$ is equal to
( where $C$ is the constant of integration)

A

\frac{e^{2 x}}{13} (2 sin 3 x - 3 cos 3 x) + C

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B

\frac{e^{2 x}}{9} (2 sin 3 x - 3 cos 3 x) + C

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C

\frac{e^{2 x}}{13} (3 sin 3 x - 2 cos 3 x) + C

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D

\frac{e^{2 x}}{13} (2 sin 3 x + 3 cos 3 x) + C

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Solution

The correct option is A $\frac{e^{2 x}}{13} (2 sin 3 x - 3 cos 3 x) + C$
$I = \int e^{2 x} sin 3 x d x \Rightarrow I = \frac{e^{2 x}}{2^{2} + 3^{2}} (2 sin 3 x - 3 cos 3 x) + C (∵ \int e^{a x} sin b x = \frac{e^{a x}}{a^{2} + b^{2}} (a sin b x - b cos b x)) ∴ I = \frac{e^{2 x}}{13} (2 sin 3 x - 3 cos 3 x) + C$

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