Transwiki:積分公式

${\displaystyle {\rm {\int kdx=kx+C}}}$
${\displaystyle {\rm {\int x^{a}dx={\frac {1}{a+1}}x^{a+1}+C}}}$
${\displaystyle {\rm {\int e^{x}dx=e^{x}+C}}}$
${\displaystyle {\rm {\int e^{ax}dx={\frac {1}{a}}e^{ax}+C}}}$
${\displaystyle {\rm {\int a^{x}dx={\frac {1}{\ln a}}a^{x}+C}}}$
${\displaystyle {\rm {\int {\frac {1}{x}}dx=\ln x+C}}}$
${\displaystyle {\rm {\int {\frac {1}{ax+b}}dx={\frac {1}{a}}\ln(ax+b)+C}}}$
${\displaystyle {\rm {\int \sin xdx=-\cos x+C}}}$
${\displaystyle {\rm {\int \sin axdx=-{\frac {1}{a}}\cos ax+C}}}$
${\displaystyle {\rm {\int \cos xdx=\sin x+C}}}$
${\displaystyle {\rm {\int \cos axdx={\frac {1}{a}}\sin ax+C}}}$
${\displaystyle {\rm {I_{n}=\int _{0}^{\frac {\pi }{2}}\sin ^{n}xdx=\int _{0}^{\frac {\pi }{2}}\cos ^{n}xdx={\frac {n-1}{n}}I_{n-2}}}}$
${\displaystyle {\rm {\int {\frac {dx}{\cos ^{2}x}}=\int \sec ^{2}xdx=\tan x+C}}}$
${\displaystyle {\rm {\int {\frac {dx}{\sin ^{2}x}}=\int \csc ^{2}xdx=-\cot x+C}}}$
${\displaystyle {\rm {\int \sec x\cdot \tan xdx=\sec x+C}}}$
${\displaystyle {\rm {\int \csc x\cdot \cot xdx=-\csc x+C}}}$
${\displaystyle {\rm {\int \tan xdx=-\ln |\cos x|+C}}}$
${\displaystyle {\rm {\int \cot xdx=\ln |\sin x|+C}}}$
${\displaystyle {\rm {\int \sec xdx=\ln |\sec(x)+\tan(x)|+C}}}$
${\displaystyle {\rm {\int \csc xdx=\ln |\csc(x)-\cot(x)|+C}}}$
${\displaystyle {\rm {\int \sinh xdx=\cosh x+C}}}$
${\displaystyle {\rm {\int \cosh xdx=\sinh x+C}}}$
${\displaystyle {\rm {\int {\frac {dx}{\sqrt {1-x^{2}}}}=\arcsin x+C}}}$
${\displaystyle {\rm {\int {\frac {dx}{\sqrt {a^{2}-x^{2}}}}=\arcsin {\frac {x}{a}}+C}}}$
${\displaystyle {\rm {\int {\frac {dx}{1+x^{2}}}=\arctan x+C}}}$
${\displaystyle {\rm {\int {\frac {dx}{a^{2}+x^{2}}}={\frac {1}{a}}\arctan {\frac {x}{a}}+C}}}$
${\displaystyle {\rm {\int {\frac {dx}{a^{2}-x^{2}}}={\frac {1}{2a}}\ln {\frac {a+x}{a-x}}+C}}}$
${\displaystyle {\rm {\int {\frac {dx}{x^{2}-a^{2}}}={\frac {1}{2a}}\ln |{\frac {x-a}{x+a}}|+C}}}$
${\displaystyle {\rm {\int {\frac {dx}{\sqrt {x^{2}\pm a^{2}}}}=\ln(x+{\sqrt {x^{2}\pm a^{2}}})+C}}}$
${\displaystyle {\rm {\int {\sqrt {x^{2}+a^{2}}}dx={\frac {x}{2}}{\sqrt {x^{2}+a^{2}}}+{\frac {a^{2}}{2}}\ln(x+{\sqrt {x^{2}+a^{2}}})+C}}}$
${\displaystyle {\rm {\int {\sqrt {x^{2}-a^{2}}}dx={\frac {x}{2}}{\sqrt {x^{2}-a^{2}}}-{\frac {a^{2}}{2}}\ln |x+{\sqrt {x^{2}-a^{2}}}|+C}}}$
${\displaystyle {\rm {\int {\sqrt {a^{2}-x^{2}}}dx={\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}+{\frac {a^{2}}{2}}\arcsin {\frac {x}{a}}+C}}}$