## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{1}(x+1) \cdot \ln ^{2}(x+1) d x
$$
## Solution

$$
\int_{0}^{1}(x+1) \cdot \ln ^{2}(x+1) d x=
$$

Let's denote:

$$
\begin{aligned}
& u=\ln ^{2}(x+1) ; d u=2 \ln (x+1) \cdot \frac{1}{x+1} \cdot d x \\
& d v=(x+1) \cdot d x ; v=\frac{(x+1)^{2}}{2}
\end{aligned}
$$

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

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

Let's denote:

$$
\begin{aligned}
& u=\ln (x+1) ; d u=\frac{1}{x+1} d x \\
& d v=(x+1) \cdot d x ; v=\frac{(x+1)^{2}}{2}
\end{aligned}
$$

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

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

$$
\begin{aligned}
& =2 \ln ^{2} 2-\left(\ln (1+1) \cdot(1+1)-\ln (0+1) \cdot(0+1)-\frac{1}{2} \int_{0}^{1}(x+1) \cdot d x\right)= \\
& =2 \ln ^{2} 2-\left(2 \ln 2-0 \cdot 1-\left.\frac{1}{2} \cdot \frac{(x+1)^{2}}{2}\right|_{0} ^{1}\right)=2 \ln ^{2} 2-\left(2 \ln 2-\frac{(1+1)^{2}}{4}+\frac{(0+1)^{2}}{4}\right)= \\
& =2 \ln ^{2} 2-\left(2 \ln 2-1+\frac{1}{4}\right)=2 \ln ^{2} 2-2 \ln 2+\frac{3}{4}
\end{aligned}
$$

Source — «http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\�_\�\� \%D1\%83\%D0\%B7\%D0\%BD\%D0\%B5\%D1\%86\%D0\%BE\%D0\%B2_\%D0\%98\%D0\%BD\%D1\%82\%D0\%B5 $\% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B} \_2-24$ "

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

Ukrainian Banner Network

- Last modified: 21:33, 27 February 2009.
- Content is available under CC-BY-SA 3.0.

Created by GeeTeatoo

## Problem Kuznetsov Integrals 2-25

## Material from PlusPi