Основные интегралы

1. \(\displaystyle \int 0 \, dx = C\)
2. \(\displaystyle \int 1 \, dx = x + C\)
3. \(\displaystyle \int x^n \, dx = \dfrac{x^{n+1}}{n+1} + C, \quad n \neq -1, x > 0\)
4. \(\displaystyle \int \dfrac{dx}{x} = \ln|x| + C\)
5. \(\displaystyle \int a^x \, dx = \dfrac{a^x}{\ln a} + C\)
6. \(\displaystyle \int e^x \, dx = e^x + C\)
7. \(\displaystyle \int \sin x \, dx = -\cos x + C\)
8. \(\displaystyle \int \cos x \, dx = \sin x + C\)
9. \(\displaystyle \int \dfrac{dx}{\sin^2 x} = -\operatorname{ctg} x + C\)
10. \(\displaystyle \int \dfrac{dx}{\cos^2 x} = \operatorname{tg} x + C\)
11. \(\displaystyle \int \dfrac{dx}{\sqrt{a^2 - x^2}} = \arcsin \dfrac{x}{a} + C, \quad |x| < a\)
12. \(\displaystyle \int \dfrac{dx}{a^2 + x^2} = \dfrac{1}{a} \operatorname{arctg} \dfrac{x}{a} + C\)
13. Высокий логарифм: \(\displaystyle \int \dfrac{dx}{a^2 - x^2} = \dfrac{1}{2a} \ln \left| \dfrac{a + x}{a - x} \right| + C, \quad |x| \neq a\)
14. Длинный логарифм: \(\displaystyle \int \dfrac{dx}{\sqrt{x^2 \pm a^2}} = \ln\left(x + \sqrt{x^2 \pm a^2}\right) + C\)