Evaluate : e^x [x+2 (x+3)^2 ] d x - Vidyadip

Question

Evaluate : $\int e^x\left[\frac{x+2}{(x+3)^2}\right] \cdot d x$

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Answer

$
\begin{aligned}
\mathrm{I} & =\int e^x\left[\frac{x+3-1}{(x+3)^2}\right] \cdot d x \\
& =\int e^x\left[\frac{x+3}{(x+3)^2}+\frac{-1}{(x+3)^2}\right] \cdot d x \\
& =\int e^x\left[\frac{1}{x+3}+\frac{-1}{(x+3)^2}\right] \cdot d x \\
& \therefore f(x)=\frac{1}{x+3} \Rightarrow f^{\prime}(x)=\frac{-1}{(x+3)^2} \\
& \therefore \int e^x\left[f(x)+f^{\prime}(x)\right] \cdot d x=e^x \cdot f(x)+c \\
& =e^x \cdot\left(\frac{1}{x+3}\right)+c \\
& =\frac{e^x}{x+3}+c \\
\therefore \quad & \int e^x\left[\frac{x+2}{(x+3)^2}\right] \cdot d x=\frac{e^x}{x+3}+c
\end{aligned}
$

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