## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{1} \frac{x^{3}}{x^{2}+1} d x
$$
## Solution

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

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B} \_4-14$ " Categories: Kuznetsov's Problem Book Integrals Problem 4 | Integrals

- Last edited: 16:49, March 3, 2009.
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Created by GeeTeatoo

## Problem Kuznetsov Integrals 4-15

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