## Problem Statement

Calculate the indefinite integral:

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

$$
\begin{aligned}
& \int \frac{1-\sqrt{x}}{\sqrt{x} \cdot(x+1)} d x=\int \frac{1}{\sqrt{x} \cdot(x+1)} d x-\int \frac{1}{x+1} d x= \\
& =2 \cdot \int \frac{1}{x+1} d(\sqrt{x})-\ln |x+1|=2 \arctan \sqrt{x}-\ln |x+1|+C
\end{aligned}
$$

Source — "http://pluspi.org/wiki/index.php/%D0%97%D0%B0%D0%B4%D0%B0%D1%87%D0%BD%D0%B8%D0%BA_%D0%9A%D1%83%D0%B7%D0%BD%D0%B5%D1%86%D0%BE%D0%B2%D0%B0_%D0%98%D0%BD%D1%82%D0%B5%D0%B3%D1%80%D0%B0%D0%BB%D1%8B_%D0%97%D0%B0%D0%B4%D0%B0%D1%87%D0%B0_33-31"

Categories: Kuznetsov's Problem Book Integrals Problem 3 | Integrals

- Last edited on this page: 14:19, 3 March 2009.
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