Question:
Evaluate the following integrals:
$\int e^{x}\left(\frac{1}{x^{2}}-\frac{2}{x^{3}}\right) d x$
Solution:
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,
$\int \mathrm{x}^{\mathrm{n}} \mathrm{dx}=\frac{\mathrm{x}^{\mathrm{n}+1}}{\mathrm{n}+1}$
$=e^{x} x^{-2}+2 \int e^{x} x^{-3} d x-2 \int e^{x} x^{-3} d x$
$=\frac{e^{x}}{x^{2}}+c$
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.