How do you find the indefinite integral of $\int e^{2 x} sin (7 x) d x$ $int e^(2x)sin(7x)dx$ ?

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How do you find the indefinite integral of $\int e^{2 x} sin (7 x) d x$ $int e^(2x)sin(7x)dx$ ?

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Answer 1 · 2015-09-20T14:16:22

$\frac{1}{53} e^{2 x} (2 sin 7 x - 7 cos 7 x) + C$ $1/53e^(2x)(2sin 7x-7cos 7x)+C$

Explanation:

Integration by parts:

$\int u d v = u v - \int v d u$ $intudv=uv-intvdu$

$I = \int e^{2 x} sin 7 x d x$ $I=inte^(2x)sin 7xdx$

$e^{2 x} = u \Rightarrow 2 e^{2 x} d x = d u$ $e^(2x)=u => 2e^(2x)dx=du$
$d v = sin 7 x d x \Rightarrow v = \int sin 7 x d x = - \frac{1}{7} cos 7 x$ $dv=sin 7xdx => v=intsin 7xdx=-1/7cos 7x$

$I = - \frac{1}{7} e^{2 x} cos 7 x + \frac{2}{7} \int e^{2 x} cos 7 x d x$ $I=-1/7e^(2x)cos 7x+2/7inte^(2x)cos 7xdx$

Again:

$e^{2 x} = u \Rightarrow 2 e^{2 x} d x = d u$ $e^(2x)=u => 2e^(2x)dx=du$

$d v = cos 7 x d x \Rightarrow v = \int cos 7 x d x = \frac{1}{7} sin 7 x$ $dv=cos 7xdx => v=intcos 7xdx=1/7sin 7x$

$I = - \frac{1}{7} e^{2 x} cos 7 x + \frac{2}{7} [\frac{1}{7} e^{2 x} sin 7 x - \frac{2}{7} \int e^{2 x} sin 7 x d x]$ $I=-1/7e^(2x)cos 7x+2/7[1/7e^(2x)sin7x-2/7inte^(2x)sin7xdx]$

$I = - \frac{1}{7} e^{2 x} cos 7 x + \frac{2}{49} e^{2 x} sin 7 x - \frac{4}{49} \int e^{2 x} sin 7 x d x$ $I=-1/7e^(2x)cos 7x+2/49e^(2x)sin7x-4/49inte^(2x)sin7xdx$

$I = - \frac{1}{7} e^{2 x} cos 7 x + \frac{2}{49} e^{2 x} sin 7 x - \frac{4}{49} I$ $I=-1/7e^(2x)cos 7x+2/49e^(2x)sin7x-4/49I$

$I + \frac{4}{49} I = - \frac{1}{7} e^{2 x} cos 7 x + \frac{2}{49} e^{2 x} sin 7 x$ $I+4/49I=-1/7e^(2x)cos 7x+2/49e^(2x)sin7x$

$\frac{53}{49} I = - \frac{1}{7} e^{2 x} cos 7 x + \frac{2}{49} e^{2 x} sin 7 x$ $53/49I=-1/7e^(2x)cos 7x+2/49e^(2x)sin7x$

$I = - \frac{7}{53} e^{2 x} cos 7 x + \frac{2}{53} e^{2 x} sin 7 x + C$ $I=-7/53e^(2x)cos 7x+2/53e^(2x)sin7x+C$

$I = \frac{1}{53} e^{2 x} (2 sin 7 x - 7 cos 7 x) + C$ $I=1/53e^(2x)(2sin 7x-7cos 7x)+C$

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How do you find the indefinite integral of $\int e^{2 x} sin (7 x) d x$ $int e^(2x)sin(7x)dx$ ?

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Explanation:

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How do you find the indefinite integral of $\int e^{2 x} sin (7 x) d x$ $int e^(2x)sin(7x)dx$ ?