Problems on Techniques of Integration

Use the Integration by Parts for definite integrals. 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_0^1 xe^{x} dx = [x e^{x}]_0^1 - \int_0^1 e^{x} dx \end{displaymath}

or

\begin{displaymath}\int_0^1 x e^{x} dx = [x e^{x}]_0^1 - [e^{x}]_0^1 = e - (e-1) = 1\;.\end{displaymath}

Detailed Answer.


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