Math.it

Integrali indefiniti fondamentali

${\displaystyle \int f^{\prime }\left(x\right) dx=f\left(x\right)+c}$

${\displaystyle \int a dx=ax+c}$

${\displaystyle \int x^n dx=\frac{x^{n+1}}{n+1}+c}$ , con ${\displaystyle n\ne -1}$

${\displaystyle \int \frac{1}{x} dx=\log \left|x\right|+c}$

${\displaystyle \int \frac{1}{x^2} dx=- \frac{1}{x}+c}$

${\displaystyle \int \sqrt{x} dx=\frac{2}{3}\sqrt{x^3}+c}$

${\displaystyle \int \frac{1}{\sqrt{x}} dx=2\sqrt{x}+c}$

${\displaystyle \int \sin x dx=-\cos x+c}$

${\displaystyle \int \cos x dx=\sin x+c}$

${\displaystyle \int (1+{\tan }^{2}x) dx=\int \frac{1}{{\cos }^2 x}dx=\tan x+c}$

${\displaystyle \int (1+{\cot }^{2}x) dx=\int \frac{1}{{\sin }^2 x}dx=-\cot x+c}$

${\displaystyle \int \sinh x dx=\cosh x+c}$

${\displaystyle \int \cosh x dx=\sinh x+c}$

${\displaystyle \int e^x dx=e^x+c}$

${\displaystyle \int e^{kx} dx=\frac{e^{kx}}{k}+c}$

${\displaystyle \int a^x dx=\frac{a^x}{{\log }_{e}a}+c}$

Integrali notevoli

${\displaystyle \int \frac{1}{\sin x} dx=\{\begin{array}{l}\log \left|\tan \frac{x}{2}\right|+c\\ \log \left|\csc x-\cot x\right|+c\end{array}}$

${\displaystyle \int \frac{1}{\cos x} dx=\{\begin{array}{l}\log \left|\tan (\frac{x}{2}+\frac{\pi }{4})\right|+c\\ \log \left|\sec x+\tan x\right|+c\end{array}}$

${\displaystyle \int \frac{1}{\sqrt{1-x^2}} dx=\{\begin{array}{l}arc\sin x+c\\ -arc\cos x+c\end{array}}$

${\displaystyle \int \frac{-1}{\sqrt{1-x^2}} dx=\{\begin{array}{l}arc\cos x+c\\ -arc\sin x+c\end{array}}$

${\displaystyle \int \frac{1}{1+x^2} dx=arc\tan x+c}$

${\displaystyle \int \frac{1}{1-x^2} dx=\frac{1}{2}\log \left|\frac{1+x}{1-x}\right|+c}$

${\displaystyle \int \frac{1}{\sqrt{x^2-1}} dx=\log \left|x+\sqrt{x^2-1}\right|+c}$

${\displaystyle \int \frac{1}{\sqrt{1+x^2}} dx=\{\begin{array}{l}arc\sinh x+c\\ \log (x+\sqrt{1+x^2})+c\end{array}}$

${\displaystyle \int \frac{1}{\sqrt{x^2\pm a^2}} dx=\log \left|x+\sqrt{x^2\pm a^2}\right|+c}$

${\displaystyle \int \sqrt{(x^2\pm a^2)} dx=\frac{x}{2}\sqrt{x^2\pm a^2}\pm \frac{a^2}{2}\log (x+\sqrt{x^2\pm a^2})+c}$

${\displaystyle \int \sqrt{(a^2-x^2)} dx=\frac{1}{2}(a^{2}arc\sin \frac{x}{a}+x\sqrt{a^2-x^2})+c}$

${\displaystyle \int {\sin }^{2}x dx=\frac{1}{2}(x-\sin x\cos x)+c}$

${\displaystyle \int {\cos }^{2}x dx=\frac{1}{2}(x+\sin x\cos x)+c}$

${\displaystyle \int \frac{1}{{\cosh }^2 x} dx=\int (1-{\tanh }^2 x)dx+c=\tanh x+c}$

${\displaystyle \int \frac{1}{{\sinh }^2 x} dx=-\coth x+c}$

${\displaystyle \int f^n\left(x\right)\cdot f^{\prime }\left(x\right) dx=\frac{f^{n+1}\left(x\right)}{n+1}+c}$

${\displaystyle \int \frac{f^{\prime }\left(x\right)}{f\left(x\right)} dx=\log \left|f\left(x\right)\right|+c}$

${\displaystyle \int f^{\prime }\left(x\right)\cdot \cos f\left(x\right) dx=\sin f\left(x\right)+c}$

${\displaystyle \int f^{\prime }\left(x\right)\cdot \sin f\left(x\right) dx=-\cos f\left(x\right)+c}$

${\displaystyle \int e^{f\left(x\right)}\cdot f^{\prime }\left(x\right) dx=e^{f\left(x\right)}+c}$

${\displaystyle \int a^{f\left(x\right)}\cdot f^{\prime }\left(x\right) dx=\frac{a^{f\left(x\right)}}{{\log }_{e}a}+c}$

${\displaystyle \int \frac{f^{\prime }\left(x\right)}{\sqrt{1-f^2\left(x\right)}} dx=\{\begin{array}{l}arc\sin f\left(x\right)+c\\ -arc\cos f\left(x\right)+c\end{array}}$

${\displaystyle \int \frac{f^{\prime }\left(x\right)}{1+f^2\left(x\right)} dx=arc\tan f\left(x\right)+c}$