## Problem Statement

Calculate the definite integral:

$$
\int_{1}^{\sqrt{3}} \frac{d x}{\sqrt{\left(1+x^{2}\right)^{3}}}
$$
## Solution

$$
\int_{1}^{\sqrt{3}} \frac{d x}{\sqrt{\left(1+x^{2}\right)^{3}}}=
$$

Substitution:

$$
\begin{aligned}
& x=\operatorname{tg} t \Rightarrow d x=\frac{d t}{\cos ^{2} t} \\
& x=1 \Rightarrow t=\operatorname{arctg} 1=\frac{\pi}{4} \\
& x=\sqrt{3} \Rightarrow t=\operatorname{arctg} \sqrt{3}=\frac{\pi}{3}
\end{aligned}
$$

We get:

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

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 $\% \mathrm{D} 1 \% 82 \% \mathrm{D} 0 \% \mathrm{~B} 5 \% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+12-21$ » Categories: Kuznetsov's Problem Book Integrals Problem 12 | Integrals

- Last edited: 15:35, 5 July 2009.
- Content is available under CC-BY-SA 3.0.

Kuznetsov's Problem Book Integrals 12-21 - PlusPi

Created by GeeTeatoo

Creste

Proof $\qquad$

![](https://cdn.mathpix.com/cropped/2024_05_22_543e00be326595f7ce31g-37.jpg?height=34&width=54&top_left_y=29&top_left_x=1999)
佂

![](https://cdn.mathpix.com/cropped/2024_05_22_543e00be326595f7ce31g-37.jpg?height=43&width=23&top_left_y=400&top_left_x=2033)

![](https://cdn.mathpix.com/cropped/2024_05_22_543e00be326595f7ce31g-37.jpg?height=390&width=42&top_left_y=488&top_left_x=2025)

一

佂

$-$

佂

佂

![](https://cdn.mathpix.com/cropped/2024_05_22_543e00be326595f7ce31g-37.jpg?height=91&width=74&top_left_y=2790&top_left_x=2022)

## Kuznetsov's Problem Book Integrals 12-22

## Material from PlusPi