"Evaluate the following integrals. e2xcos⁡(3x)dx" was answered by students like you

Wierzycaz

Answered question

2021-10-03

Evaluate the following integrals.

e^{2 x} \cos (3 x) d x

Answer & Explanation

Elberte

Skilled2021-10-04Added 95 answers

Here,
$e^{2 x} \cos (3 x) d x = e^{2 x} \frac{\sin 3 x}{3} - \int \frac{\sin 3 x}{3} \times 2 e^{2 x} d x$
(By the method of by-parts)
$= e^{2 x} \frac{\sin 3 x}{3} - \frac{2}{3} \int e^{2 x} \sin 3 x d x$
(Again, by the method of by-parts)
$= e^{2 x} \frac{\sin 3 x}{3} - \frac{2}{3} [e^{2 x} \frac{(- \cos 3 x)}{3} - \int \frac{(- \cos 3 x)}{3} \times \frac{d}{d x} (e^{2 x}) d x]$
$= e^{2 x} \frac{\sin 3 x}{3} - \frac{2}{3} [e^{2 x} \frac{(- \cos 3 x)}{3} \mp \frac{2}{3} \int e^{2 x} \cos 3 x d x]$
$= e^{2 x} \frac{\sin 3 x}{3} + \frac{2}{9} e^{2 x} \cos 3 x - \frac{4}{9} \int e^{2 x} \cos 3 x d x$
$\Rightarrow \int e^{2 x} \cos 3 x d x + \frac{4}{9} \int e^{2 x} \cos 3 x d x = e^{2 x} \frac{\sin 3 x}{3} + \frac{2}{9} e^{2 x} \cos 3 x$
$\Rightarrow \frac{13}{9} \int e^{2 x} \cos 3 x d x = e^{2 x} \frac{\sin 3 x}{3} + \frac{2}{9} e^{2 x} \cos 3 x$

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