Example 10. Find the integral $\int \sqrt{8+2 x-x^{2}} d x$.
Solution. This is also an integral of the form (4.25). By transforming the quadratic trinomial and applying formula (4.27), we get

$$
\begin{gathered}
\int \sqrt{8+2 x-x^{2}} d x=\int \sqrt{-\left(x^{2}-2 x-8\right)}= \\
=\int \sqrt{-\left(x^{2}-2 x+1-1-8\right)} d x=\int \sqrt{-\left(x^{2}-2 x+1\right)+9} d x= \\
=\int \sqrt{3^{2}-(x-1)^{2}} d(x-1)=\frac{x-1}{2} \sqrt{3^{2}-(x-1)^{2}}+ \\
+\frac{3^{2}}{2} \arcsin \frac{x-1}{3}+C=\frac{x-1}{2} \sqrt{8+2 x-x^{2}}+ \\
+\frac{9}{2} \arcsin \frac{x-1}{3}+C
\end{gathered}
$$

## Problems

Find the integrals:

1. $\int \frac{d x}{x^{2}+36}$.
2. $\int \frac{d x}{2 x^{2}+17}$.
3. $\int \frac{d x}{3 x^{2}-10}$.
4. $\int \frac{d x}{4 x^{2}+10 x-24}$.
5. $\int \frac{d x}{4 x^{2}-5 x+2}$.
6. $\int \frac{d x}{x^{2}-x-1}$.
7. $\int \frac{x-1}{x^{2}-x-1} d x$.
8. $\int \frac{d x}{\sqrt{3 x^{2}-6 x+12}}$.
9. $\int \frac{d x}{\sqrt{2+3 x-2 x^{2}}}$.
10. $\int \frac{d x}{\sqrt{x^{2}+2 x}}$.
11. $\int \sqrt{x^{2}+4 x+13} d x$.
12. $\int \sqrt{5+4 x-x^{2}} d x$.

## Answers

1. $\frac{1}{6} \operatorname{arctg} \frac{x}{6}+$ C. 2. $\frac{1}{\sqrt{34}} \operatorname{arctg} \sqrt{\frac{2}{17}} x+$ C. 3. $\frac{1}{2 \sqrt{30}} \ln \left|\frac{\sqrt{3} x-\sqrt{10}}{\sqrt{3} x+\sqrt{10}}\right|+$
$+C$.

$$
\text { 4. } \frac{1}{22} \ln \left|\frac{2 x-3}{2 x+8}\right|+C
$$

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

$+\sqrt{x^{2}+2 x} \mid+C . \quad$ 11. $\frac{x+2}{2} \sqrt{x^{2}+4 x+13}+\frac{9}{2} \ln \left|x+2+\sqrt{x^{2}+4 x+13}\right|$.

12. $\frac{x-2}{2} \sqrt{5+4 x-x^{2}}+\frac{9}{2} \arcsin \frac{x-2}{3}+C$.

## § 4.6. Integration of Rational Functions

The indefinite integral of a whole rational function (polynomial) is found directly:

$$
\int\left(a_{0}+a_{1} x+\cdots+a_{n} x^{n}\right) d x=a_{0} \int d x+a_{1} \int x d x+\cdots+a_{n} \int x^{n} d x
$$

The integration of a fractional rational function, after isolating the whole part, reduces to the integration of a proper rational fraction

$$
\frac{P(x)}{Q(x)}
$$

where $P(x)$ and $Q(x)$ are polynomials, and the degree of $P(x)$ is lower than the degree of $Q(x)$.

If the polynomial $Q(x)$ has real distinct roots $a, b, \ldots, l$ of multiplicities $\alpha, \beta, \ldots, \lambda$ and complex conjugate roots $\alpha_{k} \pm i \beta_{k}(k=1,2, \ldots, s)$ of multiplicities $\mathrm{v}_{1}, \mathrm{v}_{2}, \ldots, \mathrm{v}_{s}$, i.e., $Q(x)=(x-a)^{\alpha}(x-b)^{\beta} \ldots(x-l)^{\lambda}\left(x^{2}+p_{1} x+\right.$ $\left.+q_{1}\right)^{v_{1}}\left(x^{2}+p_{2} x+q_{2}\right)^{v_{2}} \ldots\left(x^{2}+p_{s} x+q_{s}\right)^{v_{s}}$, then the following decomposition of the fraction (4.28) into partial fractions is valid:

$$
\begin{gathered}
\frac{P(x)}{Q(x)}=\frac{A_{1}}{(x-a)}+\frac{A_{2}}{(x-a)^{2}}+\cdots+\frac{A_{\alpha}}{(x-a)^{\alpha}}+\frac{B_{1}}{(x-b)}+\frac{B_{2}}{(x-b)^{2}}+\cdots+ \\
+\frac{B_{\beta}}{(x-b)^{\beta}}+\cdots+\frac{L_{1}}{(x-l)}+\frac{L_{2}}{(x-l)^{2}}+\cdots+\frac{L_{\lambda}}{(x-l)^{\lambda}}+\frac{C_{1} x+D_{1}}{x^{2}+p_{1} x+q_{1}}+ \\
+\frac{C_{2} x+D_{2}}{\left(x^{2}+p_{1} x+q_{1}\right)^{2}}+\cdots+\frac{C_{v_{1}} x+D_{v_{1}}}{\left(x^{2}+p_{1} x+q_{1}\right)^{v_{1}}}+\cdots+\frac{M_{1} x+N_{1}}{x^{2}+p_{s} x+q_{s}}+ \\
\quad+\frac{M_{2} x+N_{2}}{\left(x^{2}+p_{s} x+q_{s}\right)^{2}}+\cdots+\frac{M_{v_{s}} x+N_{v_{s}}}{\left(x^{2}+p_{s} x+q_{s}\right)_{s}}
\end{gathered}
$$

The constants $\quad A_{1}, A_{2}, \ldots, A_{\alpha}, \quad B_{1}, \ldots, B_{\beta}, \quad L_{1}, \ldots, L_{\lambda}, \quad C_{1}$, $D_{1}, \ldots, M_{v_{s}}, N_{\mathrm{v}_{s}}$ are found using the method of undetermined coefficients.

After decomposing into partial fractions, the integration of a proper rational fraction reduces to finding integrals of the form:

$$
\begin{gathered}
I_{1}=\int \frac{d x}{(x-a)^{n}} \\
I_{2}=\int \frac{M x+N}{\left(x^{2}+p x+q\right)^{m}} d x
\end{gathered}
$$

Integral (4.30) is found using formula (4.5'), if $n=2,3, \ldots$, or using formula (4. $6^{\prime}$ ), if $n=1$.

Integral (4.31) for $m=1$ is an integral of the form (4.21), for $m>1$ the method of reduction is applied (see example $5 \S 4.6$ ).

Remark 1. If the polynomial $Q(x)$ has a real root $b$ of multiplicity $\beta$, then in formula (4.29) it corresponds to $\beta$ partial fractions $\frac{A_{\mu}}{(x-b)^{\mu}}(\mu=1,2, \ldots, \beta)$, each of which is a fraction whose numerator is a constant and whose denominator is the corresponding power of the difference ( $x-b$ ),
and all powers from 1 to $\beta$ are taken.

Remark 2. If the polynomial $Q(x)$ has complex conjugate roots $a_{k} \pm i b_{k}$ of multiplicity $v_{k}$, then in the decomposition (4.29) they correspond to $v_{k}$ elementary fractions of the form

$$
\frac{M_{\mu} x+N_{\mu}}{\left(x^{2}+p_{k} x+q_{k}\right)^{\mu}}\left(\mu=1,2, \ldots, v_{k}\right)
$$

the numerator of each of which is a linear function (not a constant, as in the previous case), and the denominator is the power of the corresponding quadratic trinomial, i.e., the trinomial

$$
x^{2}+p_{k} x+q_{k}=\left[x-\left(a_{k}+i b_{k}\right)\right]\left[x-\