Example 11. Find the integral $\int \frac{d x}{\sqrt{1-9 x^{2}}}$.
Solution.

$$
\int \frac{d x}{\sqrt{1-9 x^{2}}}=\int \frac{d x}{\sqrt{1-(3 x)^{2}}}=\frac{1}{3} \int \frac{d(3 x)}{\sqrt{1-(3 x)^{2}}}=\frac{1}{3} \arcsin 3 x+C
$$

## Problems

Using the simplest transformations of the differential and the table of integrals $\left(9.5^{\prime}\right)-\left(9.15^{\prime}\right)$, find the indefinite integrals:

1. $\int(3 x-5)^{2} d x$.
2. $\int \sqrt{2 x+3} d x$.
3. $\int \sqrt{x^{2}-4} x d x$.
4. $\int e^{-\frac{x}{3}} d x$.
5. $\int\left(\sin \frac{x}{3}+\cos \frac{x}{3}\right) d x$.
6. $\int \frac{x d x}{x^{2}+1}$.
7. $\int \frac{d x}{\sin ^{2} \frac{x}{2}}$.
8. $\int \frac{d x}{1+\frac{x^{2}}{4}}$.
9. $\int \frac{d x}{\sqrt{1-\frac{x^{2}}{9}}}$.
10. $\int \frac{d x}{4 x^{2}-1}$.
11. $\int \operatorname{ctg} x d x$.
12. $\int \frac{2 x+6}{x^{2}+6 x+14} d x$.

Answers

1. $\frac{1}{9}(3 x-5)^{3}+C$.
2. $\frac{1}{3}(2 x+3)^{\frac{3}{2}}+C$.
3. $\frac{1}{3}\left(x^{2}-4\right)^{\frac{3}{2}}+C$.
4. $-3 e^{-\frac{x}{3}}+$ C. $5.3\left(\sin \frac{x}{3}-\cos \frac{x}{3}\right)+$ C. 6. $\frac{1}{2} \ln \left(x^{2}+1\right)+$ C. 7. $-2 \operatorname{ctg} \frac{x}{2}+$

$+C . \quad$ 8. $2 \operatorname{arctg} \frac{x}{2}+C . \quad$ 9. $3 \arcsin \frac{x}{3}+C . \quad$ 10. $\frac{1}{4} \ln \left|\frac{2 x-1}{2 x+1}\right|+C$.

11. $\ln |\sin x|+C .12 \cdot \ln \left(x^{2}+6 x+14\right)+C$.

## § 4.3. Method of Substitution

Integration by introducing a new variable (method of substitution) is based on the formula

$$
\int f(x) d x=\int f[\varphi(t)] \varphi^{\prime}(t) d t
$$

where $x=\varphi(t)$ is a differentiable function of the variable $t$.