Utente:AlfaOmega08/Sandbox

Integrali immediati

$\int dx=x+C$

$\int {\frac {1}{x}}\,dx=ln\left|x\right|+C$

$\int x^{2}\,dx={\frac {x^{3}}{3}}+C$

$\int x^{n}\,dx={\frac {x^{n+1}}{n+1}}+C$

$\int a^{x}\,dx={\frac {a^{x}}{ln\,a}}+C$

$\int e^{x}\,dx=e^{x}+C$

$\int \sin x\,dx=-\cos x+C$

$\int \cos x\,dx=\sin x+C$

Integrali complessi

$\int {\frac {f^{\prime }(x)}{f(x)}}\,dx=ln\left|f(x)\right|+C$

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

$\int -{\frac {1}{\sqrt {1-x^{2}}}}\,dx=\arccos x+C$

$\int {\frac {1}{1+x^{2}}}\,dx=\arctan x+C$

$\int {\frac {f^{\prime }(x)}{1+f^{2}(x)}}\,dx=\arctan f(x)+C$

$\int {\frac {1}{a^{2}+x^{2}}}\,dx={\frac {1}{a}}\arctan {\frac {x}{a}}+C$

Formule utili

$\int n\,f(x)\,dx=n\int f(x)\,dx$

$\int f(x)+g(x)\,dx=\int f(x)\,dx+\int g(x)\,dx$

$\int f'(g(x))\,g'(x)\,dx=f(g(x))+C$

$\int f^{\prime }(x)\,g(x)\,dx=f(x)g(x)-\int f(x)\,g^{\prime }(x)\,dx$

$\int lnx\,dx=\int 1\,lnx\,dx=x\,lnx-\int x\,{\frac {1}{x}}\,dx=x\,lnx-\int dx=x\,lnx-x$

$\int log_{b}\,x\,dx=x\,log_{b}\,x-x\,log_{b}\,e$

Altro

Area: $\int _{b}^{a}f(x)\,dx$

Volume: $\pi \int _{b}^{a}f^{2}(x)\,dx$

Lunghezza: $\int _{b}^{a}{\sqrt {1+(f^{\prime }(x))^{2}}}\,dx$

Superficie di rotazione: $2\pi \int _{b}^{a}f(x){\sqrt {1+(f^{\prime }(x))^{2}}}\,dx$

$2^{0}+2^{1}+2^{2}+2^{3}+\ldots +2^{63}=2^{64}-1$

$\sum _{n=0}^{K}2^{n}=n^{K+1}-1$

Derivate

$f(x)$	$f^{\prime }(x)$
$k$	$0$
$nx$	$n$
$x^{2}$	$2x$
$x^{n}$	$nx^{n-1}$
$lnx$	${\frac {1}{x}}$
${\frac {1}{x}}$	$-{\frac {1}{x^{2}}}$
$e^{x}$	$e^{x}$
$e^{f(x)}$	$e^{f(x)}f^{\prime }(x)$
$a^{f(x)}$	$a^{f(x)}ln\,a\,f^{\prime }(x)$
$g(x)+h(x)$	$g^{\prime }(x)+h^{\prime }(x)$
$g(h(x))$	$g^{\prime }(h(x))\,h^{\prime }(x)$
$g(x)+h(x)$	$g^{\prime }(x)+h^{\prime }(x)$

Formule trigonometriche

$\sin(\alpha +\beta )=\sin \alpha \,\cos \beta +\cos \alpha \,\sin \beta$

$\cos(\alpha +\beta )=\cos \alpha \,\cos \beta -\sin \alpha \,\sin \beta$

$sin(2\alpha )=2\sin \alpha \,\cos \alpha$

$cos(2\alpha )=\cos ^{2}\alpha -\sin ^{2}\alpha =1-2\sin ^{2}(\alpha )$

Teorema di Carnot: $a=b+c-2\,b\,c\,\sin \gamma$. Dove a, b, c sono lati di un triangolo qualunque.

Geometria analitica

Circonferenza

$x^{2}+y^{2}+ax+by+c=0$

$a=-2x_{0}\qquad b=-2y_{0}\qquad c=x_{0}^{2}+y_{0}^{2}-r^{2}$

Parabola (solo se con asse parallelo ad uno degli assi)

$ax^{2}+bx+c=0$

$\Delta =b^{2}-4ac$

Asse di simmetria: $x=-{\frac {b}{2a}}$

Vertice: $\left(-{\frac {b}{2a}};-{\frac {\Delta }{4a}}\right)$

Fuoco: $\left(-{\frac {b}{2a}};{\frac {1-\Delta }{4a}}\right)$

Direttrice: <math>y = -\frac{1 + \Delta}{4a}