Неопределенный интеграл

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 \(\int f(x)dx=F(x)+C\)
где \(F(x) \) – первообразная функции \(f(x)\) на промежутке, \(C\) – произвольная постоянная

Основные свойства

1. \(\left(\int f(x)dx\right)’=f(x)\)
2. \(d\int f(x)dx=f(x)dx\)
3. \(\int F'(x)dx=F(x)+C\)
4. \(\int dF(x) = F(x)+C\)
5. Если \(\int f(x)dx=F(x)+C\), то \(\int f(ax+b)dx=\displaystyle\frac{1}{a}F(ax+b)+C\)
6. \(\int (\alpha f(x)+\beta g(x))dx = \alpha\int f(x)dx+\beta\int g(x)dx\)

Формула интегрирования по частям

\(\int fdg = f\cdot g-\int gdf\)

Таблица неопределенных интегралов

1. \(\int 0\cdot dx = C\)
2. \(\int dx =\int 1\cdot dx = x + C\)
3. \(\int xdx = \displaystyle\frac{x^2}{2}+C\)
4. \(\int x^2dx = \displaystyle\frac{x^3}{3}+C\)
5. \(\int x^ndx=\displaystyle\frac{x^{n+1}}{n+1}+C\) при  \(n\ne -1\)
6. \(\int\displaystyle\frac{dx}{x}=\ln|x|+C\)
7. \(\int\sin xdx=-\cos x+C\)
8. \(\int \cos xdx=\sin x+C\)
9. \(\int \displaystyle\frac{dx}{\cos^2x}=\mathrm{tg} x +C\)
10. \(\int \displaystyle\frac{dx}{\sin^2x}=-\mathrm{ctg} x +C\)
11. \(\int e^xdx=e^x+C\)
12. \(\int a^xdx = \displaystyle\frac{a^x}{\ln a}+C\)
13. \(\int \mathrm{tg} xdx=-\ln|\cos x|+C\)
14. \(\int \mathrm{ctg} xdx=\ln|\sin x|+C\)
15. \(\int\displaystyle\frac{1}{\sqrt{1-x^2}}dx=\arcsin x+C\)
16. \(\int\displaystyle\frac{1}{1+x^2}dx=\mathrm{arctg} x+C\)
17. \(\int\ln{x}dx=x\ln{x}-x+C\)
18. \(\int\mathrm{sh} xdx=\mathrm{ch} x+C\)
19. \(\int\mathrm{ch} xdx=\mathrm{sh} x+C\)
20. \(\int \displaystyle\frac{dx}{\mathrm{sh}^2x}=-\mathrm{cth} x +C\)
21. \(\int \displaystyle\frac{dx}{\mathrm{ch}^2x}=\mathrm{th} x +C\)
22. \(\int\displaystyle\frac{1}{a^2+x^2}dx=\frac{1}{a}\mathrm{arctg}\frac{x}{a}+C\), \(a\ne0\)
23. \(\int\displaystyle\frac{1}{a^2-x^2}dx=\frac{1}{2a}\ln\left|\displaystyle\frac{a+x}{a-x}\right|+C\), \(a\ne0\)
24. \(\int\displaystyle\frac{1}{\sqrt{a^2-x^2}}dx=\arcsin\displaystyle\frac{x}{a}+C\), \(a\ne0\)
25. \(\int\displaystyle\frac{1}{\sqrt{x^2+a^2}}dx=\ln|x+\sqrt{x^2+a^2}|+C\), \(a\ne0\)
26. \(\int\displaystyle\frac{1}{\sqrt{x^2-a^2}}dx=\ln|x+\sqrt{x^2-a^2}|+C\), \(a\ne0\)
27. \(\int\sqrt{x^2+a}dx=\displaystyle\frac{x}{2}\sqrt{x^2+a}+\frac{a}{2}\ln|x+\sqrt{x^2+a}|+C\), \(a\ne0\)
28. \(\int\sqrt{a^2-x^2}dx=\displaystyle\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\arcsin\frac{x}{a}+C\), \(a\ne0\)