Categoria: Formulario

{\color{blue}1)}\int f'(x)dx=f(x)\\
{\color{blue}2)}\int \alpha\  dx=\alpha x+c\\
{\color{blue}3)}\int x^n dx=\frac{x^{n+1}}{n+1}+c\text{ se n}\neq-1\\
{\color{blue}4)}\int\frac1x dx=\ln|x|+c\\
{\color{blue}5)}\int e^xdx=e^x+c\\
{\color{blue}6)}\int e^{\alpha x}dx=\frac{e^{\alpha x}}{\alpha}+c\\
{\color{blue}7)}\int a^x dx=\frac {a^x}{\ln a}+c\\
{\color{blue}8)}\int\cos x dx=\sin x+c\\
{\color{blue}9)}\int\sin xdx=-\cos x+c\\
{\color{blue}10)}\int(1+\tg^2x) dx=\int\frac 1{ \cos^2 x}dx=\tg x+c\\
{\color{blue}11)}\int(1+\cotg^2x) dx=\int\frac 1{ \sin^2 x}dx=-\cotg x+c\\
{\color{blue}12)}\int\frac 1{1+x^2}dx=\arctg x +c\\
{\color{blue}13)}\int\frac1{\sqrt{1-x^2} }dx=\arcsin x+c\\
{\color{blue}14)}\int\frac{-1}{\sqrt{1-x^2} }dx=\arccos x+c\\
{\color{blue}15)}\int \sinh xdx=\cosh x+c\\
{\color{blue}16)}\int \cosh xdx=\sinh x+c

{\color{blue}1)}\int f'(g(x))\cdot g'(x)dx=f(g(x))+c\\
{\color{blue}2)}\int [f(x)]^n\cdot f'(x) dx=\frac{[f(x)]^{n+1}}{n+1}+c\text{ se n}\neq-1\\
{\color{blue}3)}\int\frac{f'(x)}{f(x)} dx=\ln|f(x)|+c\\
{\color{blue}4)}\int e^{f(x)}\cdot f'(x)dx=e^{f(x)}+c\\
{\color{blue}5)}\int a^{f(x)}\cdot f'(x) dx=\frac {a^{f(x)}}{\ln a}+c\\
{\color{blue}6)}\int\cos f(x)\cdot f'(x)dx=\sin f(x)+c\\
{\color{blue}7)}\int\sin f(x)\cdot f'(x)dx=-\cos f(x)+c\\
{\color{blue}8)}\int\frac 1{ \cos^2 f(x)}\cdot f'(x)dx=\tg f(x)+c\\
{\color{blue}9)}\int\frac 1{ \sin^2 f(x)}\cdot f'(x)dx=-\cotg f(x)+c\\
{\color{blue}10)}\int\frac 1{1+[f(x)]^2}\cdot f'(x)dx=\arctg f(x) +c\\
{\color{blue}11)}\int\frac1{\sqrt{1-[f(x)]^2} }\cdot f'(x)dx=\arcsin f(x)+c\\
{\color{blue}12)}\int\frac{-1}{\sqrt{1-[f(x)]^2} }\cdot f'(x)dx=\arccos f(x)+c\\
{\color{blue}13)}\int \sinh f(x)\cdot f'(x)dx=\cosh f(x)+c\\
{\color{blue}14)}\int \cosh f(x)\cdot f'(x)dx=\sinh f(x)+c

{\color{blue}1)}\int k\cdot f(x)dx=k\cdot \int f(x)dx\\
{\color{blue}2)}\int[f_1(x)+f_2(x)+\dots+f_n(x)]dx=\int f_1(x)dx+\int f_2(x)dx+\dots+\int f_n(x)dx

\text{pongo x=g(t), da cui otteniamo dx=g'(t)dt e di conseguenza}\\
\int f(x)dx=\int f[g(t)]\cdot g'(t)dt

\int f'(x)\cdot g(x)dx=f(x)\cdot g(x)-\int f(x)\cdot g'(x)dx

\int\arctg xdx

{\color{blue}1)} f(x)=k, \text{con k costante}\Rightarrow f'(x)=0\\
{\color{blue}2)}f(x)=x^n, n\in \R \Rightarrow
f'(x)=nx^{n-1}\\
{\color{blue}2.a)}f(x)=x \Rightarrow f'(x)=1\\
{\color{blue}2.b)}f(x)=\frac1x=x^{-1}\Rightarrow f'(x)=-(x)^{-2}=-\frac1{x^2}\\
{\color{blue}2.c)}f(x)=\sqrt x=x^{\frac12}\Rightarrow f'(x)=\frac12x^{-\frac12}=\frac1{2\sqrt x}\\
{\color{blue}3)}f(x)=\log_ax\Rightarrow f'(x)=\frac1x\log_ae=\frac{1}{x\ln a}\\
{\color{blue}3.a)}f(x)=\ln x \Rightarrow f'(x)=\frac1x\\
{\color{blue}4)} f(x)=a^x\Rightarrow f'(x)=a^x\ln a\\
{\color{blue}4.a)}f(x)=e^x\Rightarrow f'(x)=e^x\\
{\color{blue}5)}f(x)=|x|\Rightarrow f'(x)=\frac{|x|}{x}\\
{\color{blue}6)}f(x)=\sin x\Rightarrow f'(x)= \cos x \\
{\color{blue}7)}f(x)=\cos x\Rightarrow f'(x)=-\sin x\\
{\color{blue}8)}f(x)=\tg x\Rightarrow f'(x)=\frac1{\cos^2 x}=1+\tg^2x\\
{\color{blue}9)}f(x)=\arcsin x\Rightarrow f'(x)=\frac{1}{\sqrt{1-x^2}}\\
{\color{blue}10)}f(x)=\arccos x\Rightarrow f'(x)=-\frac{1}{\sqrt{1-x^2}}\\
{\color{blue}11)}f(x)=\arctg x\Rightarrow f'(x)=\frac{1}{1+x^2}

{\color{blue}1)} \text{Derivata di una costante per una funzione: } D[k\cdot f(x)]=k\cdot f'(x)\\
{\color{blue}2)}\text{Derivata di somma di funzioni: } D[f(x)+g(x)]=f'(x)+g'(x)\\
{\color{blue}3)}\text{Derivata di prodotto di funzioni: }D[f(x)\cdot(x)]=f'(x)\cdot g(x)+f(x)\cdot g'(x)\\
{\color{blue}4)}\text{Derivata di quoziente di funzioni: }D\Bigg[\frac{f(x)}{g(x)}\Bigg]=\frac{f'(x)\cdot g(x)-f(x)\cdot g'(x)}{[g(x)]^2} \text{ con g(x)}\neq0\\
{\color{blue}5)}\text{Derivata di funzione composta: }D[f(g(x))]=f'[g(x)]\cdot g'(x)

{\color{blue}1)} (1+x)^\alpha=\sum_{n=0}^{\infty} \binom{\alpha}{n}x^n=1+\alpha x +\frac{\alpha(\alpha-1)(\alpha-2)x^2}{2}+\frac{\alpha(\alpha-1)(\alpha-2)(\alpha-3)x^3}{6}+...\\
+ \binom{\alpha}{n}x^n+o(x^n) \text{ per}|x|<1, \text{ dove} \binom{\alpha}{n}=\frac{\alpha(\alpha-1)(\alpha-2)(\alpha-3)...(\alpha-n+1)}{n!}\\

{\color{blue}1.a)}\frac{1}{1+x}=(1+x)^{-1}=1-x+\frac{-1(-1-1)x^2}{2}+\frac{-1(-1-1)(-1-2)x^3}{6}+\\
+\frac{-1(-1-1)(-1-2)(-1-3)x^4}{24}-...+ \binom{-1}{n}x^n+o(x^n)=\\
=1-x+x^2-x^3+x^4-...+ (-1)^nx^n+o(x^n)\\
{\color{blue}1.b)}\sqrt{1+x}=(1+x)^{1/2}=1+\frac12x-\frac18x^2+\frac{1}{16}x^3-\frac{5}{128}x^4+...+\binom{1/2}{n}x^n+o(x^n)

\\
{\color{blue}2)} e^x=\sum_{n=0}^\infty \frac{x^n}{n!}=1+x+\frac{x^2}2+\frac{x^3}6+\frac{x^4}{24}+....+\frac{x^n}{n!}+o(x^n) \forall x\in\R\\
{\color{blue}3)}\ln(1+x)=\sum_{n=1}^\infty (-1)^{n+1}\frac{x^n}{n}=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+...+\frac{(-1)^{n+1}x^n}{n}+o(x^n) \text{ per} |x|<1\\
{\color{blue}4)}\sin(x)=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}x^{2n+1}=x-\frac{x^3}{6}+\frac{x^5}{120}-\frac{x^7}{5040}+...+ \frac{(-1)^n}{(2n+1)!}x^{2n+1}+o(x^{2n+1})\forall x\in\R\\
{\color{blue}5)}\cos(x)=\sum_{n=0}^{\infty}\frac{(-1)^n}{2n!}x^{2n}=1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}+...+ \frac{(-1)^n}{2n!}x^{2n}+o(x^{2n})\forall x\in\R\\