## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{1 / \sqrt{2}} \frac{d x}{\left(1-x^{2}\right) \sqrt{1-x^{2}}}
$$
## Solution

$$
\int_{0}^{1 / \sqrt{2}} \frac{d x}{\left(1-x^{2}\right) \sqrt{1-x^{2}}}=
$$

Substitution:

$$
\begin{aligned}
& x=\sin t ; d x=\cos t d t \\
& x=0 \Rightarrow t=\arcsin 0=0 \\
& x=\frac{1}{\sqrt{2}} \Rightarrow t=\arcsin \frac{1}{\sqrt{2}}=\frac{\pi}{4}
\end{aligned}
$$

We get:

$$
\begin{aligned}
& =\int_{0}^{\pi / 4} \frac{\cos t d t}{\sqrt{\left(1-\sin ^{2} t\right)^{3}}}=\int_{0}^{\pi / 4} \frac{\cos t d t}{\sqrt{\left(1-\sin ^{2} t\right)^{3}}}=\int_{0}^{\pi / 4} \frac{\cos t d t}{\cos ^{3} t}=\int_{0}^{\pi / 4} \frac{d t}{\cos ^{2} t}= \\
& =\left.\operatorname{tg} t\right|_{0} ^{\pi / 4}=\operatorname{tg} \frac{\pi}{4}-\operatorname{tg} 0=1
\end{aligned}
$$

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+12-29$ » Categories: Kuznetsov's Problem Book Integrals Problem 12 | Integrals

Ukrainian Banner Network

- Last edited on this page: 16:55, 5 July 2009.
- Content is available under CC-BY-SA 3.0.

Created by Geeteatoo

## Problem Kuznetsov Integrals 12-30

## Material from PlusPi