EQUAÇÕES GRACELI [3]

Uma tábua de integrais^[1] (ou tabela de integrais) é uma lista que relaciona funções a famílias de antiderivadas apropriadas. Associada às propriedades de integração, tais tabelas são ferramentas de auxílio no cálculo de integrais. Este artigo contém uma tabela de integração para funções comumente utilizadas. Ao longo do texto, $a,c\in \mathbb {R}$ são constantes dadas e $C$ denota uma constante indeterminada. As fórmulas estão apresentadas sem referência explícita do conjunto para a qual sejam válidas. Mais informações sobre elas, bem como suas demonstrações, podem ser encontradas em livros-texto de cálculo^[2]^[3]^[4] e de compêndios de matemática.^[5]^[6]^[7]^[8]^[9]

Propriedades das Integrais Indefinidas

$\int cf(x)\,dx=c\int f(x)\,dx$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int [f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int f'(x)g(x)\,dx=f(x)g(x)-\int f(x)g'(x)\,dx$

Integrais Indefinidas de Funções Simples

Funções Racionais

$\int x^{n}\,dx={\frac {x^{n+1}}{n+1}}+C\qquad {\mbox{ para }}n\neq -1$
$\int {\frac {1}{x}}\,dx=\ln {\left|x\right|}+C$ ^[10] $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int {\frac {1}{a^{2}+x^{2}}}\,dx={\frac {1}{a}}{\mbox{arc tg }}{\frac {x}{a}}+C$
$\int {\frac {1}{a^{2}-x^{2}}}dx={\frac {1}{2a}}\ln \left|{\frac {a+x}{a-x}}\right|+C$ ^[11] $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =

Logaritmos

$\int \log _{a}x\,dx=x\log _{a}x-{\frac {x}{\ln a}}+C$
$\int \ln x\,dx=x(\ln x-1)+C$

Funções Exponenciais

$\int a^{x}\,dx={\frac {a^{x}}{\ln {a}}}+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int e^{x}\,dx=e^{x}+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =

Funções Irracionais

$\int {1 \over {\sqrt {a^{2}-x^{2}}}}\,dx={\mbox{arc sen }}{\frac {x}{a}}+C$
$\int {-1 \over {\sqrt {a^{2}-x^{2}}}}\,dx=\arccos {\frac {x}{a}}+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int {\frac {1}{x{\sqrt {x^{2}-a^{2}}}}}dx={\frac {1}{a}}{\mbox{arc sec }}\left|{\frac {x}{a}}\right|+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int {\frac {1}{\sqrt {a^{2}+x^{2}}}}dx=\ln |x+{\sqrt {x^{2}+a^{2}}}|+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int {\frac {1}{\sqrt {x^{2}-a^{2}}}}dx=\ln |x+{\sqrt {x^{2}-a^{2}}}|+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int {\frac {1}{x{\sqrt {a^{2}-x^{2}}}}}dx=-{\frac {1}{a}}\ln \left|{\frac {a+{\sqrt {a^{2}-x^{2}}}}{x}}\right|+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int {\frac {1}{x{\sqrt {a^{2}+x^{2}}}}}dx=-{\frac {1}{a}}\ln \left|{\frac {a+{\sqrt {a^{2}+x^{2}}}}{x}}\right|+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =

Funções Trigonométricas

$\int \cos {x}\,dx=\operatorname {sen} {x}+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int {\mbox{sen }}{x}\,dx=-\cos {x}+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int {\mbox{tg }}{x}\,dx=-\ln {\left|\cos {x}\right|}+C$ ^[12] $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int {\mbox{cossec }}{x}\,dx=\ln {\left|{\mbox{cossec }}{x}-{\mbox{cotg }}{x}\right|}+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int \sec {x}\,dx=\ln {\left|\sec {x}+{\mbox{tg }}{x}\right|}+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int {\mbox{cotg }}{x}\,dx=\ln {\left|{\mbox{sen }}{x}\right|}+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int \sec {x}{\mbox{tg }}{x}\,dx=\sec {x}+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int {\mbox{cossec }}{x}{\mbox{cotg }}{x}\,dx=-{\mbox{cossec }}{x}+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int \sec ^{2}x\,dx={\mbox{tg }}x+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int {\mbox{cossec}}^{2}x\,dx=-{\mbox{cotg }}x+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int {\mbox{sen}}^{2}x\,dx={\frac {1}{2}}(x-{\mbox{sen }}x\cos x)+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int \cos ^{2}x\,dx={\frac {1}{2}}(x+{\mbox{sen }}x\cos x)+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =

Funções Hiperbólicas

$\int {\mbox{senh }}x\,dx=\cosh x+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int \cosh x\,dx={\mbox{senh }}x+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int {\mbox{tgh }}x\,dx=\ln(\cosh x)+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int {\mbox{cossech}}\,x\,dx=\ln \left|{\mbox{tgh }}{x \over 2}\right|+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int {\mbox{sech}}\,x\,dx={\mbox{arctg }}({\mbox{senh }}x)+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int {\mbox{cotgh }}x\,dx=\ln |{\mbox{senh }}x|+C$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =

Integrais Impróprias^[^{carece de fontes]}

$\int _{0}^{\infty }{{\sqrt {x}}\,e^{-x}\,dx}={\frac {1}{2}}{\sqrt {\pi }}$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int _{0}^{\infty }{e^{-x^{2}}\,dx}={\frac {1}{2}}{\sqrt {\pi }}$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int _{0}^{\infty }{{\frac {x}{e^{x}-1}}\,dx}={\frac {\pi ^{2}}{6}}$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int _{0}^{\infty }{{\frac {x^{3}}{e^{x}-1}}\,dx}={\frac {\pi ^{4}}{15}}$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int _{0}^{\infty }{\frac {\sin(x)}{x}}\,dx={\frac {\pi }{2}}$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
$\int _{-\infty }^{\infty }{e^{-x^{2}}\,dx}={\sqrt {\pi }}$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =

Funções Especiais

Função gama: $\Gamma (z)=\int _{0}^{\infty }x^{z-1}\,e^{-x}\,dx$ ^[13] $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
Função erro: ${\text{erf}}(x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}dt$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
Logaritmo integral: ${\text{Li}}(x)=\int _{0}^{x}{\frac {dt}{\ln t}}$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
Integral elíptica de primeiro tipo: $F(a,\theta )=\int _{0}^{{\text{sen }}\theta }{\frac {dx}{\sqrt {(1-x^{2})(1-a^{2}x^{2})}}}$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
Seno integral: ${\text{Si}}(x)=\int _{0}^{x}{\frac {{\text{sen }}t}{t}}dt$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
Cosseno integral: ${\text{Ci}}(x)=-\int _{x}^{\infty }{\frac {{\text{cos}}t}{t}}dt$ $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =

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EQUAÇÕES GRACELI [3]

Propriedades das Integrais Indefinidas

Integrais Indefinidas de Funções Simples

Funções Racionais

Logaritmos

Funções Exponenciais

Funções Irracionais

Funções Trigonométricas

Funções Hiperbólicas

Integrais Impróprias^[^{carece de fontes]}

Funções Especiais

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Propriedades das Integrais Indefinidas

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Integrais Impróprias[carece de fontes]

Funções Especiais

Comentários

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Integrais Impróprias^[^{carece de fontes]}