x + 3 ( x + 4 ) 2 ? e x ? d x is equal to:

Question

$\int \frac{x + 3}{{(x + 4)}^{2}} e^{x} d x$ is equal to:

Smart Achievers · Accepted Answer

(1)

Ans. $e^{x} (\frac{1}{x + 4}) + C$

Explanation :

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

$\frac{x + 3}{{(x + 4)}^{2}} = \frac{A}{x + 4} + \frac{B}{{(x + 4)}^{2}}$

$\Rightarrow$ $x + 3 = A (x + 4) + B$

$\Rightarrow$ $x + 3 = A x + (4 A + B)$

$\Rightarrow$ $A = 1, 4 A + B = 3$

We have, $A = 1, B = - 1$

$∴$ $I = \int (\frac{1}{x + 4} - \frac{1}{{(x + 4)}^{2}}) e^{x} d x$

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

$= \int e^{x} (\frac{1}{x + 4} - \frac{1}{{(x + 4)}^{2}}) d x$

$= e^{x} (\frac{1}{x + 4}) + C$ $[∵ \int e^{x} {f (x) + f' (x)} d x = e^{x} f (x) + C]$

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