1166. Find the integrals:
1) $\int\left(5 x^{3}-4 x^{2}+2\right) d x$
2) $\int \frac{d x}{x-2}$
3) $\int \frac{5 d x}{(x+3)^{7}}$
4) $\int \frac{2 x-5}{x^{2}+4 x+8} d x$
Solution. 1) An entire rational function is integrated directly:

$$
\begin{gathered}
\int\left(5 x^{3}-4 x^{2}+2\right) d x=5 \int x^{3} d x-4 \int x^{2} d x+2 \int d x= \\
=\frac{5}{4} x^{4}-\frac{4}{3} x^{3}+2 x+C
\end{gathered}
$$

2) Integral of a simple fraction

$$
\int \frac{d x}{x-2}=\ln |x-2|+C \quad[(1) \S 3]
$$[^9]

3) Integral also from a simple fraction

$$
\begin{gathered}
\int \frac{5 d x}{(x+3)^{7}}=5 \int(x+3)^{-7} d(x+3)=5 \frac{(x+3)^{-6}}{-6}+C= \\
=-\frac{5}{6(x+3)^{6}}+C .
\end{gathered}
$$

4) The integrand fraction is simple because it is proper and the denominator is a quadratic trinomial with complex roots.

We apply the known transformation of a quadratic trinomial into a quadratic binomial:

$$
I=\int \frac{2 x-5}{x^{2}+4 x+8} d x=\int \frac{2 x-5}{(x+2)^{2}+4} d x=\int \frac{2 t-9}{t^{2}+4} d t
$$

where $x+2=t, x=t-2, d x=d t$

$I=\int \frac{2 t}{t^{2}+4} d t-9 \int \frac{d t}{t^{2}+4}=\ln \left(t^{2}+4\right)-\frac{9}{2} \operatorname{arctg} \frac{t}{2}+C=$ $=\ln \left(x^{2}+4 x+8\right)-\frac{9}{2} \operatorname{arctg} \frac{x+2}{2}+C$.