Problems on Techniques of Integration

Use the Integration by Parts. Set

u &=&x\\
dv &=& e^{2 x}dx\;.


du &=&dx\\
v &=& \displaystyle \frac{1}{2}e^{2x}\;.


\begin{displaymath}\int xe^{x} dx = x \frac{1}{2} e^{2 x} - \int \frac{1}{2} e^{2 x} dx \end{displaymath}


\begin{displaymath}\int x e^{x} dx = \frac{x}{2} e^{2 x} - \frac{1}{4}e^{2 x} + C = \frac{1}{4}(2x-1)e^{2x} + C\;.\end{displaymath}

Detailed Answer.

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