Example 15. Find $\int \frac{x^{3}+2}{x-1} d x$.
Solution. It is possible to convert an improper fraction into the sum of a polynomial and a proper fraction:

$$
\frac{x^{3}+2}{x-1}=\frac{x^{3}-1+3}{x-1}=\frac{x^{3}-1}{x-1}+\frac{3}{x-1}
$$

Therefore,

$$
\int \frac{x^{3}+2}{x-1} d x=\int\left(x^{2}+x+1+\frac{3}{x-1}\right) d x=\frac{x^{3}}{3}+\frac{x^{2}}{2}+x+3 \ln |x-1|+C.
$$

Note that it is not necessary to write as many integrals as there are terms.

## Exercises

Find the integrals.

1. $\int\left(\frac{1}{2} x-1\right)^{2} d x$
2. $\int(\sqrt{2} x-7)^{31} d x$
3. $\int \frac{d x}{8 x+9}$.
4. $\int \frac{d x}{(8 x-9)^{24}}$.
5. $\int 2^{7 x-1} d x$.
6. $\int e^{\sqrt{8}+3 x} d x$.
7. $\int \sin (\ln 3+5 x) d x$.
8. $\int \operatorname{tg}(2 x-\sqrt{\ln 6}) d x$.
9. $\int \frac{d x}{x^{2}+5 x+9}$.
10. $\int \frac{d x}{3 x^{2}+2 x+8}$.
11. $\int \frac{d x}{x^{2}+5 x-9}$.
12. $\int \frac{d x}{3 x^{2}+x-8}$.
13. $\int \frac{d x}{\sqrt{x^{2}+x+3}}$.
14. $\int \frac{d x}{\sqrt{x^{2}+x-3}}$.
15. $\int \frac{d x}{\sqrt{-x^{2}+x+3}}$.
16. $\int \frac{d x}{\sqrt{-2 x^{2}+x+1}}$.
17. $\int\left(\frac{1}{5} e^{x}-2\right)^{3} d x$.
18. $\int\left(2 e^{x}-e^{-x}\right)^{2} e^{-3 x} d x$.
19. $\int\left(\frac{1}{2}-\sin 2 x\right)^{2} d x$.
20. $\int \frac{\cos 2 x}{\cos ^{2} x} d x$.
21. $\int\left(2 x^{3}+5 x^{2}-7 x+3\right) d x$.
22. $\int\left(\frac{1}{\sqrt[3]{x}}+\sqrt{x}\right)^{2} d x$.
23. $\int \frac{2-\sqrt{1-x^{2}}}{\sqrt{1-x^{2}}} d x$.
24. $\int \frac{2-x^{4}}{1+x^{2}} d x$
25. $\int 3^{x} \cdot e^{3 x} d x$.
26. $\int(2 \operatorname{tg} x+3 \operatorname{ctg} x)^{2} d x$.
27. $\int\left(a+\frac{b}{x-a}\right)^{2} d x$.
28. $\int \frac{x}{(x+1)^{2}} d x$
29. $\int \frac{d x}{a+b-(a-b) x^{2}}$.
30. $\int \frac{x^{2}}{x^{2}+2} d x$.
31. $\int \frac{x^{2}+5 x+7}{x+3} d x$.
32. $\int \frac{x d x}{a+b x}$.
33. $\int \frac{x^{2}+1}{x-1} d x$.
34. $\int \frac{d x}{\sqrt{9+b x}-\sqrt{19+b x}}$.
35. $\int \sin 2 x \cdot \sin 6 x d x$.
36. $\int \cos x \cdot \cos 8 x d x$.

## Answers

1. $\frac{2}{3}\left(\frac{1}{2} x-1\right)^{3}+$ C. 2. $\frac{1}{32 \sqrt{2}}(\sqrt{2} x-7)^{32}+$ C. 3. $\frac{1}{8} \ln |8 x+9|+C$.
4. $-\frac{1}{184(8 x-9)^{23}}+C .5 \cdot \frac{1}{7 \ln 2} \cdot 2^{7 x-1}+C .6 \cdot \frac{1}{3} e^{\sqrt{8}+3 x}+C$.
5. $-\frac{1}{5} \cos (\ln 3+5 x)+$ C. 8. $-\frac{1}{2} \ln |\cos 2 x-\sqrt{\ln 6}|+C$.
6. $\frac{2}{\sqrt{11}} \operatorname{arctg} \frac{2 x+5}{\sqrt{11}}+C .10 . \frac{1}{\sqrt{23}} \operatorname{arctg} \frac{3 x+1}{\sqrt{23}}+C$.
7. $\frac{1}{\sqrt{61}} \ln \left|\frac{2 x-5-\sqrt{61}}{2 x-5+\sqrt{61}}\right|+C .12 \cdot \frac{1}{\sqrt{97}} \ln \left|\frac{6 x+1-\sqrt{97}}{6 x+1+\sqrt{97}}\right|+C$.

Hint. $3 x^{2}+x-8=\left(\sqrt{3} x+\frac{1}{2 \sqrt{3}}\right)^{2}-\left(\sqrt{\frac{97}{12}}\right)^{2}$,

$\ln \left|\frac{\sqrt{3} x+\frac{1}{2 \sqrt{3}}-\sqrt{\frac{97}{12}}}{\sqrt{3}+\frac{1}{2 \sqrt{3}}+\sqrt{\frac{97}{12}}}\right|=\ln \left|\frac{6 x+1-\sqrt{97}}{6 x+1+\sqrt{97}}\right|$.
13. $\ln \left(x+\frac{1}{2}+\sqrt{x^{2}+x+3}\right)+C .14 . \ln \left|x+\frac{1}{2}+\sqrt{x^{2}+x-3}\right|+C$.

15. $\arcsin \frac{2 x-1}{\sqrt{13}}+C$. 16. $\frac{1}{\sqrt{2}} \arcsin \frac{4 x-1}{3}+C$.
16. $\frac{1}{375} e^{3 x}-\frac{3}{25} e^{2 x}+\frac{12}{5} e^{x}-8 x+C .18 .-4 e^{-x}+\frac{4}{3} e^{-3 x}-\frac{1}{5} e^{-5 x}+C$.
17. $\frac{3}{4} x+\frac{1}{2} \cos 2 x-\frac{\sin 4 x}{8}+$ C. 20. $2 x-\operatorname{tg} x+C$.
18. $\frac{1}{2} x^{4}+\frac{5}{3} x^{3}-\frac{7}{2} x^{2}+3 x+$ C. 22. $\frac{x^{2}}{2}-\frac{12}{7} x \sqrt[6]{x}+3 \sqrt[3]{x}+C$.
19. $2 \arcsin x-x+C$. 24. $\operatorname{arctg} x+x-\frac{x^{3}}{3}+C .25 . \frac{e^{3 x} \cdot 3^{x}}{3+\ln 3}+C$.
20. $4 \operatorname{tg} x-9 \operatorname{ctg} x-x+C .27 . a^{2} x+2 a b \ln |x-a|-\frac{b^{2}}{x-a}+C$.
21. $\ln |x+1|+\frac{1}{x-1}+C .29 . \frac{1}{2 \sqrt{a^{2}-b^{2}}} \ln \left|\frac{x \sqrt{a-b}+\sqrt{a+b}}{x \sqrt{a-b}-\sqrt{a+b}}\right|+C$.
22. $x-\sqrt{2} \operatorname{arctg} \frac{x}{\sqrt{2}}+C .31 \cdot \frac{x^{2}}{2}+2 x+\ln |x+3|+C$.
23. $\frac{x}{b}-\frac{a}{b^{2}} \ln |a+b x|+C$. 33. $\frac{x^{2}}{2}+x+2 \ln |x-1|+C$.
24. $-\frac{1}{