Problems on Techniques of Integration

Use the Integration by Parts. Set

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

Then

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

So

\begin{displaymath}\int x e^{x} dx = x e^{x} - \int e^{x} dx \end{displaymath}

or

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

Detailed Answer.


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