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

Solution

$I = \int \frac{(x - 4) e^{x}}{{(x - 2)}^{3}} dx I = \int e^{x} (\frac{x - 2}{{(x - 2)}^{3}} - \frac{2}{{(x - 2)}^{3}}) dx I = \int e^{x} (\frac{1}{{(x - 2)}^{2}} - \frac{2}{{(x - 2)}^{3}}) dx$

Thus the given integral is of the form,

$I = \int e^{x} |f (x) + f' (x)| dx where, f (x) = \frac{1}{{(x - 2)}^{2}}; f' (x) = \frac{- 2}{{(x - 2)}^{3}} I = \int \frac{e^{x}}{{(x - 2)}^{2}} dx - \int \frac{2 e^{x}}{{(x - 2)}^{3}} dx = \frac{e^{x}}{{(x - 2)}^{2}} - \int \frac{e^{x} (- 2)}{{(x - 2)}^{3}} dx - \int \frac{2 e^{x}}{{(x - 2)}^{3}} dx + C So, I = \frac{e^{x}}{{(x - 2)}^{2}} + C$

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