INTEGGRAIS COM ELEMENTOS MATEMÁTICOS DE GRACELI.

Gn = número de Ancelmo Graceli = π (Pi) / 1.1 = 2.8559090........

• p = progressão.

Propriedades das Integrais Indefinidas

• ${\displaystyle \int cf(x)\,dx=c\int f(x)\,dx}$ Gn [pk  px ] / [cos [pk  py ]=
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• ${\displaystyle \int [f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx}$  Gn [pk  px ] / [cos [pk  py ]=
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• ${\displaystyle \int f'(x)g(x)\,dx=f(x)g(x)-\int f(x)g'(x)\,dx}$   Gn [pk  px ] / [cos [pk  py ]=
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Integrais Indefinidas de Funções Simples

Funções Racionais

Ver artigo principal: Lista de integrais de funções racionais

• ${\displaystyle \int x^{n}\,dx={\frac {x^{n+1}}{n+1}}+C\qquad {\mbox{ para }}n\neq -1}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int {\frac {1}{x}}\,dx=\ln {\left|x\right|}+C}$^[10]   Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int {\frac {1}{a^{2}+x^{2}}}\,dx={\frac {1}{a}}{\mbox{arc tg }}{\frac {x}{a}}+C}$  Gn [pk  px ] / [cos [pk  py ]=
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• ${\displaystyle \int {\frac {1}{a^{2}-x^{2}}}dx={\frac {1}{2a}}\ln \left|{\frac {a+x}{a-x}}\right|+C}$^[11]  Gn [pk  px ] / [cos [pk  py ]=

Logaritmos

Ver artigo principal: Lista de integrais de funções logarítmicas

• ${\displaystyle \int \log _{a}x\,dx=x\log _{a}x-{\frac {x}{\ln a}}+C}$  Gn [pk  px ] / [cos [pk  py ]=
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• ${\displaystyle \int \ln x\,dx=x(\ln x-1)+C}$   Gn [pk  px ] / [cos [pk  py ]=

Funções Exponenciais

Ver artigo principal: Lista de integrais de funções exponenciais

• ${\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\ln {a}}}+C}$ Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int e^{x}\,dx=e^{x}+C}$  Gn [pk  px ] / [cos [pk  py ]=

Funções Irracionais

Ver artigo principal: Lista de integrais de funções irracionais

• ${\displaystyle \int {1 \over {\sqrt {a^{2}-x^{2}}}}\,dx={\mbox{arc sen }}{\frac {x}{a}}+C}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int {-1 \over {\sqrt {a^{2}-x^{2}}}}\,dx=\arccos {\frac {x}{a}}+C}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int {\frac {1}{x{\sqrt {x^{2}-a^{2}}}}}dx={\frac {1}{a}}{\mbox{arc sec }}\left|{\frac {x}{a}}\right|+C}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int {\frac {1}{\sqrt {a^{2}+x^{2}}}}dx=\ln |x+{\sqrt {x^{2}+a^{2}}}|+C}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int {\frac {1}{\sqrt {x^{2}-a^{2}}}}dx=\ln |x+{\sqrt {x^{2}-a^{2}}}|+C}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int {\frac {1}{x{\sqrt {a^{2}-x^{2}}}}}dx=-{\frac {1}{a}}\ln \left|{\frac {a+{\sqrt {a^{2}-x^{2}}}}{x}}\right|+C}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int {\frac {1}{x{\sqrt {a^{2}+x^{2}}}}}dx=-{\frac {1}{a}}\ln \left|{\frac {a+{\sqrt {a^{2}+x^{2}}}}{x}}\right|+C}$  Gn [pk  px ] / [cos [pk  py ]=
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Funções Trigonométricas

Ver artigo principal: Lista de integrais de funções trigonométricas

• ${\displaystyle \int \cos {x}\,dx=\operatorname {sen} {x}+C}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int {\mbox{sen }}{x}\,dx=-\cos {x}+C}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int {\mbox{tg }}{x}\,dx=-\ln {\left|\cos {x}\right|}+C}$^[12]  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int {\mbox{cossec }}{x}\,dx=\ln {\left|{\mbox{cossec }}{x}-{\mbox{cotg }}{x}\right|}+C}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int \sec {x}\,dx=\ln {\left|\sec {x}+{\mbox{tg }}{x}\right|}+C}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int {\mbox{cotg }}{x}\,dx=\ln {\left|{\mbox{sen }}{x}\right|}+C}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int \sec {x}{\mbox{tg }}{x}\,dx=\sec {x}+C}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int {\mbox{cossec }}{x}{\mbox{cotg }}{x}\,dx=-{\mbox{cossec }}{x}+C}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int \sec ^{2}x\,dx={\mbox{tg }}x+C}$  Gn [pk  px ] / [cos [pk  py ]=
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• ${\displaystyle \int {\mbox{cossec}}^{2}x\,dx=-{\mbox{cotg }}x+C}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int {\mbox{sen}}^{2}x\,dx={\frac {1}{2}}(x-{\mbox{sen }}x\cos x)+C}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int \cos ^{2}x\,dx={\frac {1}{2}}(x+{\mbox{sen }}x\cos x)+C}$  Gn [pk  px ] / [cos [pk  py ]=
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Funções Hiperbólicas

Ver artigo principal: Lista de integrais de funções hiperbólicas

• ${\displaystyle \int {\mbox{senh }}x\,dx=\cosh x+C}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int \cosh x\,dx={\mbox{senh }}x+C}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int {\mbox{tgh }}x\,dx=\ln(\cosh x)+C}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int {\mbox{cossech}}\,x\,dx=\ln \left|{\mbox{tgh }}{x \over 2}\right|+C}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int {\mbox{sech}}\,x\,dx={\mbox{arctg }}({\mbox{senh }}x)+C}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int {\mbox{cotgh }}x\,dx=\ln |{\mbox{senh }}x|+C}$  Gn [pk  px ] / [cos [pk  py ]=

Integrais Impróprias^{[carece de fontes]}

• ${\displaystyle \int _{0}^{\infty }{{\sqrt {x}}\,e^{-x}\,dx}={\frac {1}{2}}{\sqrt {\pi }}}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int _{0}^{\infty }{e^{-x^{2}}\,dx}={\frac {1}{2}}{\sqrt {\pi }}}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int _{0}^{\infty }{{\frac {x}{e^{x}-1}}\,dx}={\frac {\pi ^{2}}{6}}}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int _{0}^{\infty }{{\frac {x^{3}}{e^{x}-1}}\,dx}={\frac {\pi ^{4}}{15}}}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int _{0}^{\infty }{\frac {\sin(x)}{x}}\,dx={\frac {\pi }{2}}}$  Gn [pk  px ] / [cos [pk  py ]=

• ${\displaystyle \int _{-\infty }^{\infty }{e^{-x^{2}}\,dx}={\sqrt {\pi }}}$  Gn [pk  px ] / [cos [pk  py ]=
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Funções Especiais

• Função gama: ${\displaystyle \Gamma (z)=\int _{0}^{\infty }x^{z-1}\,e^{-x}\,dx}$^[13]  Gn [pk  px ] / [cos [pk  py ]=
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• Função erro: ${\displaystyle {\text{erf}}(x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}dt}$  Gn [pk  px ] / [cos [pk  py ]=
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• Logaritmo integral: ${\displaystyle {\text{Li}}(x)=\int _{0}^{x}{\frac {dt}{\ln t}}}$  Gn [pk  px ] / [cos [pk  py ]=
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• Integral elíptica de primeiro tipo: ${\displaystyle F(a,\theta )=\int _{0}^{{\text{sen }}\theta }{\frac {dx}{\sqrt {(1-x^{2})(1-a^{2}x^{2})}}}}$  Gn [pk  px ] / [cos [pk  py ]=

• Seno integral: ${\displaystyle {\text{Si}}(x)=\int _{0}^{x}{\frac {{\text{sen }}t}{t}}dt}$ Gn [pk  px ] / [cos [pk  py ]=

• Cosseno integral: ${\displaystyle {\text{Ci}}(x)=-\int _{x}^{\infty }{\frac {{\text{cos}}t}{t}}dt}$  Gn [pk  px ] / [cos [pk  py ]=