## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{2} \frac{x^{4} \cdot d x}{\sqrt{\left(8-x^{2}\right)^{3}}}
$$
## Solution

$$
\int_{0}^{2} \frac{x^{4} \cdot d x}{\sqrt{\left(8-x^{2}\right)^{3}}}=\int_{0}^{2} \frac{x^{4} \cdot d x}{\left(8-x^{2}\right)^{3 / 2}}=
$$

Substitution:

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

We get:

$$
\begin{aligned}
& =\int_{0}^{\pi / 4} \frac{64 \sin ^{4} t \cdot 2 \sqrt{2} \cos t}{\sqrt{\left(8-8 \sin ^{2} t\right)^{3}}} d t=\int_{0}^{\pi / 4} \frac{64 \sin ^{4} t \cdot 2 \sqrt{2} \cos t}{16 \sqrt{2} \cos ^{3} t} d t=8 \cdot \int_{0}^{\pi / 4} \frac{\sin ^{4} t}{\cos ^{2} t} d t= \\
& =8 \cdot \int_{0}^{\pi / 4} \frac{\left(1-\cos ^{2} t\right)^{2}}{\cos ^{2} t} d t=8 \cdot \int_{0}^{\pi / 4} \frac{1-2 \cos ^{2} t+\cos ^{4} t}{\cos ^{2} t} d t= \\
& =8 \cdot \int_{0}^{\pi / 4}\left(\frac{1}{\cos ^{2} t}-2+\frac{1+\cos 2 t}{2}\right) d t=8 \cdot \int_{0}^{\pi / 4}\left(\frac{1}{\cos ^{2} t}-\frac{3}{2}+\frac{1}{2} \cos 2 t\right) d t= \\
& =\left.8\left(\operatorname{tg} t-\frac{3 t}{2}+\frac{1}{4} \sin 2 t\right)\right|_{0} ^{\pi / 4}=8\left(\operatorname{tg} \frac{\pi}{4}-\frac{3 \pi}{8}+\frac{1}{4} \sin \frac{\pi}{2}\right)-8\left(\operatorname{tg} 0-\frac{3 \cdot 0}{2}+\frac{1}{4} \sin 0\right)= \\
& =8\left(1-\frac{3 \pi}{8}+\frac{1}{4} \cdot 1\right)-8\left(2 \cdot 0-0+\frac{1}{4} \cdot 0\right)=8\left(\frac{5}{4}-\frac{3 \pi}{8}\right)=10-3 \pi
\end{aligned}
$$

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 $\% \mathrm{D} 0 \% \mathrm{~B} 3 \% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+12-23$ »

Categories: Kuznetsov's Problem Book Integrals Problem 12 | Integrals

- Last edited: 16:00, July 5, 2009.
- Content is available under CC-BY-SA 3.0.

Kuznetsov's Problem Book Integrals 12-23 — PlusPi

Created by GeeTeatoo

![](https://cdn.mathpix.com/cropped/2024_05_22_543e00be326595f7ce31g-40.jpg?height=80&width=51&top_left_y=260&top_left_x=994)

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![](https://cdn.mathpix.com/cropped/2024_05_22_543e00be326595f7ce31g-40.jpg?height=60&width=60&top_left_y=146&top_left_x=1630)

![](https://cdn.mathpix.com/cropped/2024_05_22_543e00be326595f7ce31g-40.jpg?height=63&width=31&top_left_y=157&top_left_x=1839)

![](https://cdn.mathpix.com/cropped/2024_05_22_543e00be326595f7ce31g-40.jpg?height=922&width=70&top_left_y=217&top_left_x=2012)

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## Kuznetsov's Problem Book Integrals 12-24

## Material from PlusPi