## Problem Statement

Calculate the definite integral:

$$
\int_{0}^{2 \sqrt{2}} \frac{x^{4} \cdot d x}{\left(16-x^{2}\right) \sqrt{16-x^{2}}}
$$
## Solution

$$
\int_{0}^{2 \sqrt{2}} \frac{x^{4} \cdot d x}{\left(16-x^{2}\right) \sqrt{16-x^{2}}}=\int_{0}^{2 \sqrt{2}} \frac{x^{4} \cdot d x}{\left(16-x^{2}\right)^{3 / 2}}=
$$

Substitution:

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

We get:

$$
\begin{aligned}
& =\int_{0}^{\pi / 4} \frac{4^{4} \sin ^{4} t \cdot 4 \cos t}{\sqrt{\left(16-16 \sin ^{2} t\right)^{3}}} d t=\int_{0}^{\pi / 4} \frac{4^{4} \sin ^{4} t \cdot 4 \cos t}{4^{3} \cos ^{3} t} d t=16 \cdot \int_{0}^{\pi / 4} \frac{\sin ^{4} t}{\cos ^{2} t} d t= \\
& =16 \cdot \int_{0}^{\pi / 4} \frac{\left(1-\cos ^{2} t\right)^{2}}{\cos ^{2} t} d t=16 \cdot \int_{0}^{\pi / 4} \frac{1-2 \cos ^{2} t+\cos ^{4} t}{\cos ^{2} t} d t= \\
& =16 \cdot \int_{0}^{\pi / 4}\left(\frac{1}{\cos ^{2} t}-2+\frac{1+\cos 2 t}{2}\right) d t=16 \cdot \int_{0}^{\pi / 4}\left(\frac{1}{\cos ^{2} t}-\frac{3}{2}+\frac{1}{2} \cos 2 t\right) d t= \\
& =\left.16\left(\operatorname{tg} t-\frac{3 t}{2}+\frac{1}{4} \sin 2 t\right)\right|_{0} ^{\pi / 4}=16\left(\operatorname{tg} \frac{\pi}{4}-\frac{3 \pi}{8}+\frac{1}{4} \sin \frac{\pi}{2}\right)-16\left(\operatorname{tg} 0-\frac{3 \cdot 0}{2}+\frac{1}{4} \sin 0\right)= \\
& =16\left(1-\frac{3 \pi}{8}+\frac{1}{4} \cdot 1\right)-16\left(2 \cdot 0-0+\frac{1}{4} \cdot 0\right)=16\left(\frac{5}{4}-\frac{3 \pi}{8}\right)=20-6 \pi
\end{aligned}
$$

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\%D0\%B3 $\% \mathrm{D} 1 \% 80 \% \mathrm{D} 0 \% \mathrm{~B} 0 \% \mathrm{D} 0 \% \mathrm{BB} \% \mathrm{D} 1 \% 8 \mathrm{~B}+12-19$ »

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

Ukrainian Banner Network

- Last modified: 15:19, 5 July 2009.
- Content is available under CC-BY-SA 3.0.