Evaluate: x-4 (x-2)^3 e^x d x - Vidyadip

Question

Evaluate: $\int \frac{x-4}{(x-2)^3} \cdot e^x d x$

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Answer

$\text { (c) }: \int \frac{x-4}{(x-2)^3} \cdot e^x d x=\int\left[\frac{x-2}{(x-2)^3}-\frac{2}{(x-2)^3}\right] e^x d x$
$=\int\left[\frac{1}{(x-2)^2}-\frac{2}{(x-2)^3}\right] e^x d x=\frac{e^x}{(x-2)^2}+C$
${\left[\because \int\left[f(x)+f^{\prime}(x)\right] e^x d x=e^x f(x)+C\right]}$

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