## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{\sqrt{2}} \frac{x^{4} \cdot d x}{\left(4-x^{2}\right)^{3 / 2}}
$$
## Solution

$$
\int_{0}^{\sqrt{2}} \frac{x^{4} \cdot d x}{\left(4-x^{2}\right)^{3 / 2}}=
$$

Substitution:

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

We get:

$$
\begin{aligned}
& =\int_{0}^{\pi / 4} \frac{16 \sin ^{4} t \cdot 2 \cos t}{\sqrt{\left(4-4 \sin ^{2} t\right)^{3}}} d t=\int_{0}^{\pi / 4} \frac{16 \sin ^{4} t \cdot 2 \cos t}{8 \cos ^{3} t} d t=4 \cdot \int_{0}^{\pi / 4} \frac{\sin ^{4} t}{\cos ^{2} t} d t= \\
& =4 \cdot \int_{0}^{\pi / 4} \frac{\left(1-\cos ^{2} t\right)^{2}}{\cos ^{2} t} d t=4 \cdot \int_{0}^{\pi / 4} \frac{1-2 \cos ^{2} t+\cos ^{4} t}{\cos ^{2} t} d t= \\
& =4 \cdot \int_{0}^{\pi / 4}\left(\frac{1}{\cos ^{2} t}-2+\frac{1+\cos 2 t}{2}\right) d t=4 \cdot \int_{0}^{\pi / 4}\left(\frac{1}{\cos ^{2} t}-\frac{3}{2}+\frac{1}{2} \cos 2 t\right) d t= \\
& =\left.4\left(\operatorname{tg} t-\frac{3 t}{2}+\frac{1}{4} \sin 2 t\right)\right|_{0} ^{\pi / 4}=4\left(\operatorname{tg} \frac{\pi}{4}-\frac{3 \pi}{8}+\frac{1}{4} \sin \frac{\pi}{2}\right)-4\left(\operatorname{tg} 0-\frac{3 \cdot 0}{2}+\frac{1}{4} \sin 0\right)= \\
& =4\left(1-\frac{3 \pi}{8}+\frac{1}{4} \cdot 1\right)-4\left(2 \cdot 0-0+\frac{1}{4} \cdot 0\right)=4\left(\frac{5}{4}-\frac{3 \pi}{8}\right)=5-\frac{3 \pi}{2}
\end{aligned}
$$

Source — «http://pluspi.org/wiki/index.php/\�\�\�\�\�\�\�\�\�\�\�\�_\�\� \%D1\%83\%D0\%B7\%D0\%BD\%D0\%B5\%D1\%86\%D0\%BE\%D0\%B2_\%D0\%98\%D0\%BD\%D1\%82\%D0\%B5 $\% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B} \_12-28$ »

Categories: Kuznetsov's Problem Book Integrals Problem $12 \mid$ Integrals

Ukrainian Banner Network

- Last edited: 16:50, July 5, 2009.
- Content is available under CC-BY-SA 3.0.

Problem Kuznetsov Integrals 12-28 — PlusPi

Created by GeeTeatoo

![](https://cdn.mathpix.com/cropped/2024_05_22_543e00be326595f7ce31g-50.jpg?height=80&width=51&top_left_y=260&top_left_x=994)

Table

.

佂

![](https://cdn.mathpix.com/cropped/2024_05_22_543e00be326595f7ce31g-50.jpg?height=930&width=72&top_left_y=186&top_left_x=2010)

\author{

}

.

![](https://cdn.mathpix.com/cropped/2024_05_22_543e00be326595f7ce31g-50.jpg?height=192&width=68&top_left_y=2712&top_left_x=2025)

## Problem Kuznetsov Integrals 12-29

## Material from Pluspi