Problems on Techniques of Integration

Use the Integration by Parts. Set

\begin{displaymath}\left\{\begin{array}{lll}
u &=&x\\
dv &=& e^{2 x}dx\;.
\end{array}\right.\end{displaymath}

Then

\begin{displaymath}\left\{\begin{array}{lll}
du &=&dx\\
v &=& \displaystyle \frac{1}{2}e^{2x}\;.
\end{array}\right.\end{displaymath}

So

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

or

\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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