∫x+2x+42exdx is equal to - Infinity Learn by Sri Chaitanya

A
$e^{x} (\frac{x}{x + 4}) + C$
B
$e^{x} (\frac{x + 2}{x + 4}) + C$
C
$e^{x} (\frac{x - 2}{x + 4}) + C$
D
$(\frac{2 {xe}^{x}}{x + 4}) + C$

Solution:

$\begin{array}{l} \int {(\frac{x + 2}{x + 4})}^{2} e^{x} dx = \int e^{x} [\frac{x^{2} + 4 x + 4}{(x + 4)^{2}}] dx \\ = \int e^{x} [\frac{x (x + 4)}{(x + 4)^{2}} + \frac{4}{(x + 4)^{2}}] dx \\ = \int e^{x} [\frac{x}{x + 4} + \frac{4}{(x + 4)^{2}}] dx \\ = \int \frac{e^{x} x}{(x + 4)} dx + \int \frac{4 e^{x}}{(x + 4)^{2}} dx \\ = e^{x} (\frac{x}{x + 4}) - \int \frac{4 e^{x}}{(x + 4)^{2}} dx + \int \frac{4 e^{x}}{(x + 4)^{2}} dx \\ = \frac{{xe}^{x}}{(x + 4)} + C \end{array}$

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

Solution:

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