## Problem Statement

Calculate the definite integral:

$$
\int_{1}^{3} \frac{1-\sqrt{x}}{\sqrt{x} \cdot(x+1)} d x
$$
## Solution

$$
\begin{aligned}
& \int_{1}^{3} \frac{1-\sqrt{x}}{\sqrt{x} \cdot(x+1)} d x=\int_{1}^{3} \frac{1}{\sqrt{x} \cdot(x+1)} d x-\int_{1}^{3} \frac{1}{x+1} d x= \\
& =2 \cdot \int_{1}^{3} \frac{1}{x+1} d(\sqrt{x})-\ln |x+1|_{1}^{3}=\left.2 \operatorname{arctg} \sqrt{x}\right|_{1} ^{3}-\ln |3+1|+\ln |1+1|= \\
& =2 \operatorname{arctg} \sqrt{3}-2 \operatorname{arctg} \sqrt{1}-\ln 4+\ln 2=2 \cdot \frac{\pi}{3}-2 \cdot \frac{\pi}{4}-2 \ln 2+\ln 2= \\
& =\frac{2 \pi}{3}-\frac{\pi}{2}-\ln 2=\frac{\pi}{6}-\ln 2
\end{aligned}
$$

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} \_4-16 »

Categories: Kuznetsov's Problem Book Integrals Problem 4 | Integrals

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## Problem Kuznetsov Integrals 4-17

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