Find $\int x^{2}\,e^{x}\,dx$.

Solution We use the integration by parts formula with $$u=x^{2}\qquad \hbox{and}\qquad dv=e^{x}\,dx$$

Then $$du=2x\,dx\qquad \hbox{and}\qquad v=\int e^{x}\,dx=e^{x}$$

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and $$\int x^{2} e^{x}\,dx=x^{2}e^{x}-2\int xe^{x}\,dx$$

The integral on the right is simpler than the original integral. To find it, we use integration by parts a second time. (Refer to Example 1.)

$$\begin{equation*} \int xe^{x}\,dx=xe^{x}-e^{x} \end{equation*}$$

Then $$\int x^{2}e^{x}\,dx=x^{2}e^{x}-2( xe^{x}-e^{x}) +C=e^{x}(x^{2}-2x+2)+C$$