Evaluate the following integrals:


Evaluate the following integrals:

$\int e^{x}\left(\frac{1}{x^{2}}-\frac{2}{x^{3}}\right) d x$


Let I $=\int e^{x}\left(\frac{1}{x^{2}}-\frac{2}{x^{3}}\right) d x$

$=\int e^{x} x^{-2} d x-2 \int e^{x} x^{-3} d x$

Integrating by parts

$=x^{-2} \int e^{x} d x-\int \frac{d}{d x} x^{-2} \int e^{x} d x-2 \int e^{x} x^{-3} d x$

We know that,

$\int \mathrm{x}^{\mathrm{n}} \mathrm{dx}=\frac{\mathrm{x}^{\mathrm{n}+1}}{\mathrm{n}+1}$

$=e^{x} x^{-2}+2 \int e^{x} x^{-3} d x-2 \int e^{x} x^{-3} d x$


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