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To evaluate the integral ,
we have to apply formula of integration by parts
![\int {P .Q} dx = P \int Q dx-\int [\frac{dP}{dx}\int Q dx]dx](https://tex.z-dn.net/?f=%5Cint+%7BP+.Q%7D+dx+%3D+P+%5Cint+Q+dx-%5Cint+%5B%5Cfrac%7BdP%7D%7Bdx%7D%5Cint+Q+dx%5Ddx "\int {P .Q} dx = P \int Q dx-\int [\frac{dP}{dx}\int Q dx]dx")
Here we assume P = x
![\int x.{e}^{-2x}\:dx = x\int {e}^{-2x}\:dx-\int[\frac{dx}{dx}\int\: {e}^{-2x}dx]dx\\\\\\=\frac{-x.{e}^{-2x}}{2}+\int(1)\frac{{e}^{-2x}}{2}dx\\\\\\=\frac{-x.{e}^{-2x}}{2}-\frac{{e}^{-2x}}{4}+C](https://tex.z-dn.net/?f=%5Cint+x.%7Be%7D%5E%7B-2x%7D%5C%3Adx+%3D+x%5Cint+%7Be%7D%5E%7B-2x%7D%5C%3Adx-%5Cint%5B%5Cfrac%7Bdx%7D%7Bdx%7D%5Cint%5C%3A+%7Be%7D%5E%7B-2x%7Ddx%5Ddx%5C%5C%5C%5C%5C%5C%3D%5Cfrac%7B-x.%7Be%7D%5E%7B-2x%7D%7D%7B2%7D%2B%5Cint%281%29%5Cfrac%7B%7Be%7D%5E%7B-2x%7D%7D%7B2%7Ddx%5C%5C%5C%5C%5C%5C%3D%5Cfrac%7B-x.%7Be%7D%5E%7B-2x%7D%7D%7B2%7D-%5Cfrac%7B%7Be%7D%5E%7B-2x%7D%7D%7B4%7D%2BC "\int x.{e}^{-2x}\:dx = x\int {e}^{-2x}\:dx-\int[\frac{dx}{dx}\int\: {e}^{-2x}dx]dx\\\\\\=\frac{-x.{e}^{-2x}}{2}+\int(1)\frac{{e}^{-2x}}{2}dx\\\\\\=\frac{-x.{e}^{-2x}}{2}-\frac{{e}^{-2x}}{4}+C")
Here we assume P = x
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