Example. Find the indefinite integral

$$
\int \frac{x^{2}}{\sqrt{9-x^{2}}} d x
$$
Solution.

1. To get rid of the radical, we will use the substitution $x=3 \sin t$. Then $d x=3 \cos t$ and $\sqrt{9-x^{2}}=3 \cos t$.
2. We will make a variable substitution in the indefinite integral:

$$
\int \frac{x^{2}}{\sqrt{9-x^{2}}} d x=\int \frac{9 \sin ^{2} t 3 \cos t}{\sqrt{9-9 \sin ^{2} t}} d t=9 \int \sin ^{2} t d t
$$

3. Applying the formulas for reducing the degree, we get

$9 \int \sin ^{2} t d t=\frac{9}{2} \int(1-\cos 2 t) d t=\frac{9}{2} t-\frac{9}{2} \int \cos 2 t d t=\frac{9}{2} t-\frac{9}{4} \sin 2 t+C$.

4. We return to the variable $x$, substituting $t=\arcsin (x / 3)$ :

$$
\int \frac{x^{2}}{\sqrt{9-x^{2}}} d x=\frac{9}{2} \arcsin \frac{x}{3}-\frac{9}{4} \sin \left(2 \arcsin \frac{x}{3}\right)+C
$$

Remark. The answer can be simplified by using the fact that $\sin 2 t=2 \sin t \sqrt{1-\sin ^{2} t}$ and $\sin t=x / 3$ :

$$
\int \frac{x^{2}}{\sqrt{9-x^{2}}} d x=\frac{9}{2} \arcsin \frac{x}{3}-\frac{x}{2} \sqrt{9-x^{2}}+C
$$

Answer. $\int \frac{x^{2}}{\sqrt{9-x^{2}}} d x=\frac{9}{2} \arcsin \frac{x}{3}-\frac{x}{2} \sqrt{9-x^{2}}+C$.

Conditions of the Problems. Find the indefinite integrals.

1. $\int \sqrt{4-x^{2}} d x$.
2. $\int \sqrt{2+x^{2}} d x$.
3. $\int \sqrt{x^{2}-4} d x$.
4. $\int \frac{d x}{x \sqrt{x^{2}-1}}$.
5. $\int \frac{\sqrt{9-x^{2}}}{x} d x$.
6. $\int \frac{d x}{x \sqrt{4+x^{2}}}$.
7. $\int \frac{x^{2} d x}{\sqrt{x^{2}-4}}$.
8. $\int \frac{d x}{\sqrt{\left(1-x^{2}\right)^{3}}} d x$.
9. $\int \frac{d x}{\sqrt{\left(2+x^{2}\right)^{3}}}$.
10. $\int \frac{d x}{x^{3} \sqrt{x^{2}-1}}$.

## Answers.

1. $2 \arcsin \frac{x}{2}+\frac{x}{2} \sqrt{4-x^{2}}+C$.
2. $\ln \left(x+\sqrt{2+x^{2}}\right)+\frac{x}{2} \sqrt{2+x^{2}}+C$.
3. $\frac{x}{2} \sqrt{x^{2}-4}-2 \ln \left|x+\sqrt{x^{2}-4}\right|+C$.
4. $\arccos \frac{1}{x}+C$.
5. $3 \ln \left|\frac{3-\sqrt{9-x^{2}}}{x}\right|+\sqrt{9-x^{2}}+C$.
6. $\frac{1}{2} \ln \left|\frac{\sqrt{4+x^{2}}-2}{x}\right|+C$.
7. $2 \ln \left|x+\sqrt{x^{2}-4}\right|+\frac{x}{2} \sqrt{x^{2}-4}+C$.
8. $\frac{x}{\sqrt{1-x^{2}}}+C$.
9. $\frac{x}{2 \sqrt{2+x^{2}}}+C$.
10. $\frac{1}{2}\left(\arccos \frac{1}{x}+\frac{\sqrt{x^{2}-1}}{x^{2}}\right)+C$.

### 7.10. Integration of a Differential Binomial

Problem Statement. Find the indefinite integral

$$
\int x^{m}\left(a+b x^{n}\right)^{p} d x
$$

where $m, n$ and $p$ are rational numbers.

Plan of Solution. The expression $x^{m}\left(a+b x^{n}\right)^{p} d x$ is called a differential binomial. The conditions for its integrability in terms of elementary functions were obtained by P. L. Chebyshev.

Chebyshev's Conditions. The integral (1) can be expressed through a finite combination of elementary functions in the following three cases:

1) $p$ is an integer; in this case, the substitution $x=z^{s}$, where $s$ is the least common denominator of the fractions $m$ and $n$, reduces the integral to an integral of a rational function.
2) $\frac{m+1}{n}$ is an integer; in this case, the substitution $a+b x^{n}=z^{s}$, where $s$ is the denominator of the fraction $p$, reduces the integral to an integral of a rational function.
3) $\frac{m+1}{n}+p$ is an integer; in this case, the substitution $a x^{-n}+b=$ $=z^{s}$, where $s$ is the denominator of the fraction $p$, reduces the integral to an integral of a rational function.