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Таблица на основни интеграли - Википедија

Таблица на основни интеграли

Од Википедија, слободна енциклопедија

[уреди] Основни интеграли

  • \int x^p\,dx = \frac{x^{p+1}}{p+1} + C, \,\,\,\ p \neq -1


  • \int \frac{dx}{x} = \operatorname{ln}|x| + C


  • \int a^x\,dx = \frac{a^x}{\operatorname{ln}x} + C, \,\,\,\ a>0, \,\ a \neq 1


  • \int e^x\,dx = e^x + C


  • \int \operatorname{sin}x\,dx = -\operatorname{cos}x + C


  • \int \operatorname{cos}x\,dx = \operatorname{sin}x + C


  • \int \frac{dx}{\operatorname{cos}^2 x} = \operatorname{tg}x + C


  • \int \frac{dx}{\operatorname{sin}^2 x} = -\operatorname{ctg}x + C


  • \int \frac{dx}{\sqrt{1-x^2}} = \operatorname{arcsin}x + C


  • \int \frac{dx}{1+x^2} = \operatorname{arctg}x + C


[уреди] Останати интеграли

  • \int \frac{dx}{ax + b} = \operatorname{ln}|ax+b| + C, \,\,\,\ a \neq 0


  • \int \frac{dx}{x^2 + a^2} = \frac{1}{a} \operatorname{arctg}\frac{x}{a} + C


  • \int \frac{dx}{x^2 - a^2} = \frac{1}{2a}\operatorname{ln} \left | \frac{x-a}{x+a} \right | + C


  • \int \frac{dx}{ax^2 + bx + c} = \begin{cases} \frac{1}{\sqrt{b^2-4ac}}\operatorname{ln}\left |\frac{2ax+b-\sqrt{b^2-4ac}}{2ax+b+\sqrt{b^2-4ac}}\right | + C, & b^2-4ac>0 \\ \\ \frac{1}{\sqrt{4ac-b^2}}\operatorname{arctg}\frac{2ax+b}{4ac-b^2} + C, & b^2-4ac<0 \end{cases}


  • \int \frac{dx}{\sqrt{x^2 \pm a^2}} = \operatorname{ln}\left | x+\sqrt{x^2 \pm a^2} \right | + C


  • \int \frac{dx}{\sqrt{a^2 - x^2}} = \operatorname{sgn}a \cdot \operatorname{arcsin} \frac{x}{a} + C


  • \int \frac{dx}{\sqrt{ax^2+bx+c}} = \frac{1}{\sqrt{a}}\operatorname{ln} \left | x+\frac{b}{2a} +\sqrt{x^2 + \frac{b}{a}x + \frac{c}{a}} \right | + C


  • \int \sqrt{x^2 \pm a^2}\,dx = \frac{1}{2}\left [ x\sqrt{x^2\pm a^2} + a^2 \operatorname{ln}\left | x+\sqrt{x^2 \pm a^2}\right | \right ] + C


  • \int \sqrt{a^2-x^2}\,dx = \frac{1}{2} \left [ \operatorname{sgn}a \cdot a^2 \operatorname{arcsin}\frac{x}{a} + x\sqrt{a^2-x^2} \right ] + C


  • \int \sqrt{ax^2+bx+c}\,dx = \frac{\sqrt{a}}{2} \left [ \left ( x+\frac{b}{2a} \right ) \sqrt{x^2+\frac{b}{a}x+\frac{c}{a}} + \frac{4ac-b^2}{4a^2}\operatorname{ln} \left | x+\frac{b}{2a} + \sqrt{x^2+\frac{b}{a}x+\frac{c}{a}} \right| \right ] + C
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