Problems on Techniques of Integration

Use the Integration by Parts. Set

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

Then

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

So

\begin{displaymath}\int x^2 e^x dx = x^2 e^x - \int 2 x e^x dx = x^2 e^x - 2 \int x e^x dx\;.\end{displaymath}

In order to integrate the function $x e^x$, we will need to do another 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} \;.\end{displaymath}

Hence

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

Detailed Answer.


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