CÁLCULO GRACELI SINTÉTICO [30]

$\int cf(x)\,dx=c\int f(x)\,dx$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int [f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int f'(x)g(x)\,dx=f(x)g(x)-\int f(x)g'(x)\,dx$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds

Integrais Indefinidas de Funções Simples[editar | editar código-fonte]

Funções Racionais[editar | editar código-fonte]

Ver artigo principal: Lista de integrais de funções racionais

$\int {\frac {1}{x}}\,dx=\ln {\left|x\right|}+C$ ^[10] $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int {\frac {1}{a^{2}+x^{2}}}\,dx={\frac {1}{a}}{\mbox{arc tg }}{\frac {x}{a}}+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int {\frac {1}{a^{2}-x^{2}}}dx={\frac {1}{2a}}\ln \left|{\frac {a+x}{a-x}}\right|+C$ ^[11] $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds

Logaritmos[editar | editar código-fonte]

Ver artigo principal: Lista de integrais de funções logarítmicas

$\int \log _{a}x\,dx=x\log _{a}x-{\frac {x}{\ln a}}+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int \ln x\,dx=x(\ln x-1)+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds

Funções Exponenciais[editar | editar código-fonte]

Ver artigo principal: Lista de integrais de funções exponenciais

$\int a^{x}\,dx={\frac {a^{x}}{\ln {a}}}+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int e^{x}\,dx=e^{x}+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds

Funções Irracionais[editar | editar código-fonte]

Ver artigo principal: Lista de integrais de funções irracionais

$\int {1 \over {\sqrt {a^{2}-x^{2}}}}\,dx={\mbox{arc sen }}{\frac {x}{a}}+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int {-1 \over {\sqrt {a^{2}-x^{2}}}}\,dx=\arccos {\frac {x}{a}}+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int {\frac {1}{x{\sqrt {x^{2}-a^{2}}}}}dx={\frac {1}{a}}{\mbox{arc sec }}\left|{\frac {x}{a}}\right|+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int {\frac {1}{\sqrt {a^{2}+x^{2}}}}dx=\ln |x+{\sqrt {x^{2}+a^{2}}}|+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int {\frac {1}{\sqrt {x^{2}-a^{2}}}}dx=\ln |x+{\sqrt {x^{2}-a^{2}}}|+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int {\frac {1}{x{\sqrt {a^{2}-x^{2}}}}}dx=-{\frac {1}{a}}\ln \left|{\frac {a+{\sqrt {a^{2}-x^{2}}}}{x}}\right|+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int {\frac {1}{x{\sqrt {a^{2}+x^{2}}}}}dx=-{\frac {1}{a}}\ln \left|{\frac {a+{\sqrt {a^{2}+x^{2}}}}{x}}\right|+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds

Funções Trigonométricas[editar | editar código-fonte]

Ver artigo principal: Lista de integrais de funções trigonométricas

$\int \cos {x}\,dx=\operatorname {sen} {x}+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int {\mbox{sen }}{x}\,dx=-\cos {x}+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int {\mbox{tg }}{x}\,dx=-\ln {\left|\cos {x}\right|}+C$ ^[12] $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int {\mbox{cossec }}{x}\,dx=\ln {\left|{\mbox{cossec }}{x}-{\mbox{cotg }}{x}\right|}+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int \sec {x}\,dx=\ln {\left|\sec {x}+{\mbox{tg }}{x}\right|}+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int {\mbox{cotg }}{x}\,dx=\ln {\left|{\mbox{sen }}{x}\right|}+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int \sec {x}{\mbox{tg }}{x}\,dx=\sec {x}+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int {\mbox{cossec }}{x}{\mbox{cotg }}{x}\,dx=-{\mbox{cossec }}{x}+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int \sec ^{2}x\,dx={\mbox{tg }}x+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int {\mbox{cossec}}^{2}x\,dx=-{\mbox{cotg }}x+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int {\mbox{sen}}^{2}x\,dx={\frac {1}{2}}(x-{\mbox{sen }}x\cos x)+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int \cos ^{2}x\,dx={\frac {1}{2}}(x+{\mbox{sen }}x\cos x)+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds

Funções Hiperbólicas[editar | editar código-fonte]

Ver artigo principal: Lista de integrais de funções hiperbólicas

$\int {\mbox{senh }}x\,dx=\cosh x+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int \cosh x\,dx={\mbox{senh }}x+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int {\mbox{tgh }}x\,dx=\ln(\cosh x)+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int {\mbox{cossech}}\,x\,dx=\ln \left|{\mbox{tgh }}{x \over 2}\right|+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int {\mbox{sech}}\,x\,dx={\mbox{arctg }}({\mbox{senh }}x)+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int {\mbox{cotgh }}x\,dx=\ln |{\mbox{senh }}x|+C$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds

Integrais Impróprias^[^{carece de fontes]}[editar | editar código-fonte]

$\int _{0}^{\infty }{{\sqrt {x}}\,e^{-x}\,dx}={\frac {1}{2}}{\sqrt {\pi }}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int _{0}^{\infty }{e^{-x^{2}}\,dx}={\frac {1}{2}}{\sqrt {\pi }}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int _{0}^{\infty }{{\frac {x}{e^{x}-1}}\,dx}={\frac {\pi ^{2}}{6}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int _{0}^{\infty }{{\frac {x^{3}}{e^{x}-1}}\,dx}={\frac {\pi ^{4}}{15}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int _{0}^{\infty }{\frac {\sin(x)}{x}}\,dx={\frac {\pi }{2}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
$\int _{-\infty }^{\infty }{e^{-x^{2}}\,dx}={\sqrt {\pi }}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds

Funções Especiais[editar | editar código-fonte]

Função gama: $\Gamma (z)=\int _{0}^{\infty }x^{z-1}\,e^{-x}\,dx$ ^[13] $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
Função erro: ${\text{erf}}(x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}dt$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
Logaritmo integral: ${\text{Li}}(x)=\int _{0}^{x}{\frac {dt}{\ln t}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
Integral elíptica de primeiro tipo: $F(a,\theta )=\int _{0}^{{\text{sen }}\theta }{\frac {dx}{\sqrt {(1-x^{2})(1-a^{2}x^{2})}}}$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
Seno integral: ${\text{Si}}(x)=\int _{0}^{x}{\frac {{\text{sen }}t}{t}}dt$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds
Cosseno integral: ${\text{Ci}}(x)=-\int _{x}^{\infty }{\frac {{\text{cos}}t}{t}}dt$ $\zeta (s)\ =\ \sum _{{n=1}}^{\infty }\ {\frac {1}{n^{s}}}.$ { [nx] $n!$ / [pw] } ${\sqrt {\frac {2}{\pi }}}$ + $({\frac {\pi }{2}})^{1/2}$ .+ ${\frac {\pi }{2}}t^{2}$ ds

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