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Evaluate the ...

Question

Evaluate the integral
$\int_{0}^{\infty} e^{- 4 x} cos 3 x d x$

A

\frac{3}{25}

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B

\frac{4}{25}

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C

\frac{- 1}{25}

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D

\frac{7}{25}

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Solution

The correct option is B $\frac{4}{25}$
$\int_{0}^{\infty} e^{- 4 x} c o s 3 x d x$
$I = \int e^{- 4 x} c o s 3 x d x = c o s 3 x \int e^{- 4 x} d x - \frac{3}{4} \int + s i n 3 x e^{- 4 x} d x$
$= - c o s \frac{3 x e}{4} - \frac{3}{4} \int s i n 3 x e^{- 4 x} d x$
$\int e^{- 4} s i n 3 x d x = \frac{- s i n 3 x}{4} e^{- 4 x} + \frac{3}{4} \int c o s 3 x e^{- 4 x} d x$
$\int e^{- 4 x} c o s 3 x d x = - \frac{(c o s 3 x) e^{- 4 x}}{4} - \frac{3}{4} [\frac{- s i n 3 x (e^{- 4 x})}{4} + \frac{3}{4} \int c o s 3 x e^{- 4 x}]$
$I + \frac{9}{16} I = \frac{e^{- 4 x}}{4} [- c o s 3 x + \frac{3}{4} s i n 3 x]$
$I = \frac{4}{25} [e^{- 4 x}] (- c o s 3 x - \frac{3}{4} s i n 3 x)$
$\int_{0}^{\infty} e^{- 4 x} c o s 3 x d x = \frac{4}{25} e^{- 4 x} {[- c o s 3 x + \frac{3}{4} s i n 3 x]}_{0}^{\infty}$
$= \frac{4}{25}$

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