Интеграл тоолол


Интегралын чанарууд:

$$\int dx = x + c$$

$$\int k f(x)dx = k \int f(x) dx$$

$$\int(f(x) + \varphi(x) + \xi(x) + \dots)dx = \int f(x)dx + \int \varphi(x)dx + \int \xi(x) dx + \dots$$

$$\int udv = uv – \int vdu$$

$$\int x^n dx = \frac{x^{n+1}}{n+1}+c \hspace{1cm} (n \ne -1)$$

$$\int a^x dx = \frac{a^x}{\ln a}+ c \hspace{1cm} (0<a \ne 1)$$

$$\int \frac{dx}{x}=\ln |x| + c$$

$$\int e^x dx = e^x + c$$

Тригонометрийн функцийн интегралууд:

$$\int \sin x dx = – \cos x + c$$

$$\int \cos x dx = \sin x + c$$

$$\int \frac{dx}{\sin^2 x} = – \textrm{ctg}x + c$$

$$\int \frac{dx}{\cos^2 x} = \textrm{tg}x + c$$

$$\int \frac{dx}{\sqrt{a^2 – x^2}} = \arcsin \frac{x}{a} + c$$

$$\int \frac{dx}{1 + x^2} = \textrm{arctg} x + c$$

$$\int \frac{dx}{a^2 + x^2} = \frac{1}{a} \textrm{arctg} \frac{x}{a} + c$$

$$\int \frac{dx}{x^2 – a^2} = \frac{1}{2a} \ln \big| \frac{x-a}{x+a}\big| +c \hspace{1cm} (a \ne 0)$$

$$\int \frac{dx}{x^2 – 1} = \frac{1}{2} \ln \big| \frac{x-1}{x+1} \big| + c$$

$$\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \textrm{arctg} \frac{x}{a} + c $$

$$ \int \frac{dx}{x^2 + 1} = \textrm{arctg} x + c$$

$$\int \frac{dx}{\sqrt{x^2 + k}} = \ln \big| x + \sqrt{x^2 + k} \big| + c$$