I=∫(x+3)e^x/(x+4)^2dx I=∫(x+4 1)e^x/(x+4)^2dx I=∫e^x/(x+4)dx ∫e^x/(x+4)^2dx I=∫e^x/(x+4)dx e^x[ 1/(x+4)] ∫ e^x( 1/(x+4))dx I=∫e^xdx/(x+ ...

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Special Integrals - 1

∫ x+3 e x x+...

Question

$\int \frac{(x + 3) e^{x}}{{(x + 4)}^{2}} d x =$

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Solution

$I = \int \frac{(x + 3) e^{x}}{(x + 4)^{2}} d x$
$I = \int \frac{(x + 4 - 1) e^{x}}{(x + 4)^{2}} d x$
$I = \int \frac{e^{x}}{(x + 4)} d x - \int \frac{e^{x}}{(x + 4)^{2}} d x$
$I = \int \frac{e^{x}}{(x + 4)} d x - e^{x} [\frac{- 1}{(x + 4)}] - \int e^{x} (\frac{- 1}{(x + 4)}) d x$
$I = \int \frac{e^{x} d x}{(x + 4)} + [\frac{e^{x}}{(x + 4)}] - \int \frac{e^{x}}{(x + 4)} d x$
$I = \frac{e^{x}}{x + 4} + c$

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