ИНТЕГРАЛЫ
Таблица интегралов:
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$$\int dx=x+C$$ $$\int {f}'(x)dx=f(x)+C$$ $$\int x^{\alpha} dx=\frac {x^{\alpha+1}}{\alpha+1}+C, \alpha \neq-1$$ $$\int \frac{dx}{x}=\ln \left | x \right |+C$$ $$\int a^x dx=\frac {a^x}{\ln (a)}+C$$ $$\int e^x dx= e^x +C$$ $$\int \sin(x)dx= - \cos (x) +C$$ $$\int \cos(x)dx= \sin (x) +C$$ |
$$\int \textrm{tg}(x)d x= - \ln \left | \cos (x) \right | +C$$ $$\int \textrm{ctg}(x)d x= \ln \left | \sin (x) \right | +C$$ $$\int \frac {dx}{\sin(x)}= \ln \left | \textrm{tg} \left ( \frac {x}{2} \right) \right | +C$$ $$\int \frac {dx}{\cos(x)}= \ln \left | \textrm{tg} \left ( \frac {x}{2} + \frac {\pi}{4} \right) \right | +C$$ $$\int \frac {dx}{\sin^2(x)}= - \textrm{ctg} (x)+C$$ $$\int \frac {dx}{\cos^2(x)}= \textrm{tg} (x)+C$$ $$\int \textrm{ch}(x)dx= \textrm{sh}(x)+C$$ |
$$\int \textrm{ch}(x)dx= \textrm{sh}(x)+C$$ $$\int \textrm{sh}(x)dx= \textrm{ch}(x)+C$$ $$\int \frac {dx}{\textrm{ch}^2(x)}= \textrm{th} (x)+C$$ $$\int \frac {dx}{\textrm{sh}^2(x)}= -\textrm{cth} (x)+C$$ $$\int \frac {dx}{a^2+x^2}= \frac1 {a} \textrm{arctg} \left ( \frac {x}{a} \right)+C$$ $$\int \frac {dx}{a^2-x^2}= \frac1 {2a} \ln \left | \frac {a+x}{a-x} \right |+C$$ $$\int \frac {dx}{\sqrt {a^2-x^2}}= \arcsin \left ( \frac {x}{a} \right )+C$$ |
Методы интегрирования:
- Замена переменной: [tex]\int f\left ( u(x) \right )u'(x)dx = F\left ( u(x) \right )+C, \; F'=f[/tex]
- Интегрирование по частям: [tex]\int udv =uv-\int vdu[/tex]
- Формула Ньютона- Лейбница: [tex]\int_{a}^{b}f(x)dx=\color {red}{F(b)-F(a)}=\color {blue} {F|^a_b}[/tex]
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