∫ d x = x + C {\displaystyle \int dx=x+C}
∫ 1 x d x = l n | x | + C {\displaystyle \int {\frac {1}{x}}\,dx=ln\left|x\right|+C}
∫ x 2 d x = x 3 3 + C {\displaystyle \int x^{2}\,dx={\frac {x^{3}}{3}}+C}
∫ x n d x = x n + 1 n + 1 + C {\displaystyle \int x^{n}\,dx={\frac {x^{n+1}}{n+1}}+C}
∫ a x d x = a x l n a + C {\displaystyle \int a^{x}\,dx={\frac {a^{x}}{ln\,a}}+C}
∫ e x d x = e x + C {\displaystyle \int e^{x}\,dx=e^{x}+C}
∫ sin x d x = − cos x + C {\displaystyle \int \sin x\,dx=-\cos x+C}
∫ cos x d x = sin x + C {\displaystyle \int \cos x\,dx=\sin x+C}
∫ f ′ ( x ) f ( x ) d x = l n | f ( x ) | + C {\displaystyle \int {\frac {f^{\prime }(x)}{f(x)}}\,dx=ln\left|f(x)\right|+C}
∫ 1 1 − x 2 d x = arcsin x + C {\displaystyle \int {\frac {1}{\sqrt {1-x^{2}}}}\,dx=\arcsin x+C}
∫ − 1 1 − x 2 d x = arccos x + C {\displaystyle \int -{\frac {1}{\sqrt {1-x^{2}}}}\,dx=\arccos x+C}
∫ 1 1 + x 2 d x = arctan x + C {\displaystyle \int {\frac {1}{1+x^{2}}}\,dx=\arctan x+C}
∫ f ′ ( x ) 1 + f 2 ( x ) d x = arctan f ( x ) + C {\displaystyle \int {\frac {f^{\prime }(x)}{1+f^{2}(x)}}\,dx=\arctan f(x)+C}
∫ 1 a 2 + x 2 d x = 1 a arctan x a + C {\displaystyle \int {\frac {1}{a^{2}+x^{2}}}\,dx={\frac {1}{a}}\arctan {\frac {x}{a}}+C}
∫ n f ( x ) d x = n ∫ f ( x ) d x {\displaystyle \int n\,f(x)\,dx=n\int f(x)\,dx}
∫ f ( x ) + g ( x ) d x = ∫ f ( x ) d x + ∫ g ( x ) d x {\displaystyle \int f(x)+g(x)\,dx=\int f(x)\,dx+\int g(x)\,dx}
∫ f ′ ( g ( x ) ) g ′ ( x ) d x = f ( g ( x ) ) + C {\displaystyle \int f'(g(x))\,g'(x)\,dx=f(g(x))+C}
∫ f ′ ( x ) g ( x ) d x = f ( x ) g ( x ) − ∫ f ( x ) g ′ ( x ) d x {\displaystyle \int f^{\prime }(x)\,g(x)\,dx=f(x)g(x)-\int f(x)\,g^{\prime }(x)\,dx}
∫ l n x d x = ∫ 1 l n x d x = x l n x − ∫ x 1 x d x = x l n x − ∫ d x = x l n x − x {\displaystyle \int lnx\,dx=\int 1\,lnx\,dx=x\,lnx-\int x\,{\frac {1}{x}}\,dx=x\,lnx-\int dx=x\,lnx-x}
∫ l o g b x d x = x l o g b x − x l o g b e {\displaystyle \int log_{b}\,x\,dx=x\,log_{b}\,x-x\,log_{b}\,e}
Area: ∫ b a f ( x ) d x {\displaystyle \int _{b}^{a}f(x)\,dx}
Volume: π ∫ b a f 2 ( x ) d x {\displaystyle \pi \int _{b}^{a}f^{2}(x)\,dx}
Lunghezza: ∫ b a 1 + ( f ′ ( x ) ) 2 d x {\displaystyle \int _{b}^{a}{\sqrt {1+(f^{\prime }(x))^{2}}}\,dx}
Superficie di rotazione: 2 π ∫ b a f ( x ) 1 + ( f ′ ( x ) ) 2 d x {\displaystyle 2\pi \int _{b}^{a}f(x){\sqrt {1+(f^{\prime }(x))^{2}}}\,dx}
2 0 + 2 1 + 2 2 + 2 3 + … + 2 63 = 2 64 − 1 {\displaystyle 2^{0}+2^{1}+2^{2}+2^{3}+\ldots +2^{63}=2^{64}-1}
∑ n = 0 K 2 n = n K + 1 − 1 {\displaystyle \sum _{n=0}^{K}2^{n}=n^{K+1}-1}
sin ( α + β ) = sin α cos β + cos α sin β {\displaystyle \sin(\alpha +\beta )=\sin \alpha \,\cos \beta +\cos \alpha \,\sin \beta }
cos ( α + β ) = cos α cos β − sin α sin β {\displaystyle \cos(\alpha +\beta )=\cos \alpha \,\cos \beta -\sin \alpha \,\sin \beta }
s i n ( 2 α ) = 2 sin α cos α {\displaystyle sin(2\alpha )=2\sin \alpha \,\cos \alpha }
c o s ( 2 α ) = cos 2 α − sin 2 α = 1 − 2 sin 2 ( α ) {\displaystyle cos(2\alpha )=\cos ^{2}\alpha -\sin ^{2}\alpha =1-2\sin ^{2}(\alpha )}
Teorema di Carnot: a = b + c − 2 b c sin γ {\displaystyle a=b+c-2\,b\,c\,\sin \gamma } . Dove a, b, c sono lati di un triangolo qualunque.
x 2 + y 2 + a x + b y + c = 0 {\displaystyle x^{2}+y^{2}+ax+by+c=0}
a = − 2 x 0 b = − 2 y 0 c = x 0 2 + y 0 2 − r 2 {\displaystyle a=-2x_{0}\qquad b=-2y_{0}\qquad c=x_{0}^{2}+y_{0}^{2}-r^{2}}
a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0}
Δ = b 2 − 4 a c {\displaystyle \Delta =b^{2}-4ac}
Asse di simmetria: x = − b 2 a {\displaystyle x=-{\frac {b}{2a}}}
Vertice: ( − b 2 a ; − Δ 4 a ) {\displaystyle \left(-{\frac {b}{2a}};-{\frac {\Delta }{4a}}\right)}
Fuoco: ( − b 2 a ; 1 − Δ 4 a ) {\displaystyle \left(-{\frac {b}{2a}};{\frac {1-\Delta }{4a}}\right)}
Direttrice: <math>y = -\frac{1 + \Delta}{4a}
. Dove a, b, c sono lati di un triangolo qualunque.
