## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{1} x^{2} \cdot e^{3 x} d x
$$
## Solution

$$
\int_{0}^{1} x^{2} \cdot e^{3 x} d x=
$$

Let's denote:

$$
\begin{aligned}
& u=x^{2} ; d u=2 x \cdot d x \\
& d v=e^{3 x} \cdot d x ; v=\frac{e^{3 x}}{3}
\end{aligned}
$$

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

$$
\begin{aligned}
& =\left.x^{2} \cdot \frac{e^{3 x}}{3}\right|_{0} ^{1}-\int_{0}^{1} \frac{e^{3 x}}{3} \cdot 2 x \cdot d x=1^{2} \cdot \frac{e^{3 \cdot 1}}{3}-0^{2} \cdot \frac{e^{3 \cdot 0}}{3}-\frac{2}{3} \cdot \int_{0}^{1} x \cdot e^{3 x} \cdot d x= \\
& =\frac{e^{3}}{3}-\frac{2}{3} \cdot \int_{0}^{1} x \cdot e^{3 x} \cdot d x=
\end{aligned}
$$

Let's denote:

$$
\begin{aligned}
& u=x ; d u=d x \\
& d v=e^{3 x} \cdot d x ; v=\frac{e^{3 x}}{3}
\end{aligned}
$$

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

$$
=\frac{e^{3}}{3}-\frac{2}{3} \cdot\left(\left.x \cdot \frac{e^{3 x}}{3}\right|_{0} ^{1}-\int_{0}^{1} \frac{e^{3 x}}{3} d x\right)=\frac{e^{3}}{3}-\frac{2}{3} \cdot\left(1 \cdot \frac{e^{3 \cdot 1}}{3}-0 \cdot \frac{e^{3 \cdot 0}}{3}-\frac{1}{3} \cdot \int_{0}^{1} e^{3 x} d x\right)=
$$

$$
\begin{aligned}
& =\frac{e^{3}}{3}-\frac{2}{3} \cdot\left(\frac{e^{3}}{3}-\left.\frac{1}{3} \cdot \frac{e^{3 x}}{3}\right|_{0} ^{1}\right)=\frac{e^{3}}{3}-\frac{2}{3} \cdot\left(\frac{e^{3}}{3}-\frac{e^{3 \cdot 1}}{9}+\frac{e^{3 \cdot 0}}{9}\right)= \\
& =\frac{e^{3}}{3}-\frac{2}{3} \cdot\left(\frac{3 e^{3}}{9}-\frac{e^{3}}{9}+\frac{1}{9}\right)=\frac{e^{3}}{3}-\frac{2}{3} \cdot\left(\frac{2 e^{3}}{9}+\frac{1}{9}\right)= \\
& =\frac{9 e^{3}}{27}-\frac{4 e^{3}}{27}-\frac{2}{27}=\frac{5 e^{3}-2}{27}
\end{aligned}
$$

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B} 2-30$ »

Categories: Kuznetsov's Problem Book Integrals Problem 2 | Integrals

Ukrainian Banner Network

- Last modified: 21:17, 28 February 2009.
- Content is available under CC-BY-SA 3.0.

Created by GeeTeatoo

## Problem Kuznetsov Integrals 2-31

## Material from PlusPi