## Problem Statement

Calculate the definite integral:

$$
\int_{-1}^{1} x^{2} \cdot e^{-\frac{x}{2}} d x
$$
## Solution

$$
\int_{-1}^{1} x^{2} \cdot e^{-\frac{x}{2}} d x=
$$

Let's denote:

$$
\begin{aligned}
& u=x^{2} ; d u=2 x \cdot d x \\
& d v=e^{-\frac{x}{2}} \cdot d x ; v=-2 e^{-\frac{x}{2}}
\end{aligned}
$$

We will use the integration by parts formula $\int u d v=u v-\int v d u$. We get:

$$
\begin{aligned}
& =\left.x^{2} \cdot\left(-2 e^{-\frac{x}{2}}\right)\right|_{-1} ^{1}-\int_{-1}^{1}\left(-2 e^{-\frac{x}{2}}\right) \cdot 2 x \cdot d x=1^{2} \cdot\left(-2 e^{-\frac{1}{2}}\right)-(-1)^{2} \cdot\left(-2 e^{-\frac{-1}{2}}\right)+4 \cdot \int_{-1}^{1} x \cdot e^{-\frac{x}{2}} \cdot d x= \\
& =-\frac{2}{\sqrt{e}}+2 \sqrt{e}+4 \cdot \int_{-1}^{1} x \cdot e^{-\frac{x}{2}} \cdot d x=
\end{aligned}
$$

Let's denote:

$$
\begin{aligned}
& u=x ; d u=d x \\
& d v=e^{-\frac{x}{2}} \cdot d x ; v=-2 e^{-\frac{x}{2}}
\end{aligned}
$$

We will use the integration by parts formula $\int u d v=u v-\int v d u$. We get:

$$
\begin{aligned}
& =-\frac{2}{\sqrt{e}}+2 \sqrt{e}+4 \cdot\left(\left.x \cdot\left(-2 e^{-\frac{x}{2}}\right)\right|_{-1} ^{1}-\int_{-1}^{1}\left(-2 e^{-\frac{x}{2}}\right) d x\right)= \\
& =-\frac{2}{\sqrt{e}}+2 \sqrt{e}+4 \cdot\left(1 \cdot\left(-2 e^{-\frac{1}{2}}\right)-(-1) \cdot\left(-2 e^{-\frac{-1}{2}}\right)+2 \cdot \int_{-1}^{1} e^{-\frac{x}{2}} d x\right)= \\
& =-\frac{2}{\sqrt{e}}+2 \sqrt{e}+4 \cdot\left(-\frac{2}{\sqrt{e}}-2 \sqrt{e}+\left.2 \cdot\left(-2 e^{-\frac{x}{2}}\right)\right|_{-1} ^{1}\right)= \\
& =-\frac{2}{\sqrt{e}}+2 \sqrt{e}+4 \cdot\left(-\frac{2}{\sqrt{e}}-2 \sqrt{e}-4 e^{-\frac{1}{2}}+4 e^{-\frac{-1}{2}}\right)= \\
& =-\frac{2}{\sqrt{e}}+2 \sqrt{e}+4 \cdot\left(-\frac{2}{\sqrt{e}}-2 \sqrt{e}-\frac{4}{\sqrt{e}}+4 \sqrt{e}\right)=
\end{aligned}
$$

$$
=-\frac{2}{\sqrt{e}}+2 \sqrt{e}+4 \cdot\left(-\frac{6}{\sqrt{e}}+2 \sqrt{e}\right)=-\frac{2}{\sqrt{e}}+2 \sqrt{e}-\frac{24}{\sqrt{e}}+8 \sqrt{e}=-\frac{26}{\sqrt{e}}+10 \sqrt{e}
$$

Source — «http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\� \%D0\%9A \%D1\%83\%D0\%B7\%D0\%BD\%D0\%B5\%D1\%86\%D0\%BE\%D0\%B2_\%D0\%98\%D0\%BD\%D1\%82\%D0\%B5\%D0\%B3 $\% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B} \_2-29$ »

Categories: Kuznetsov's Integral Problems 2 | Integrals

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## Problem Kuznetsov Integrals 2-30

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