∫kdx=kx+C{\displaystyle {\rm {\int kdx=kx+C}}}∫xadx=1a+1xa+1+C{\displaystyle {\rm {\int x^{a}dx={\frac {1}{a+1}}x^{a+1}+C}}}∫exdx=ex+C{\displaystyle {\rm {\int e^{x}dx=e^{x}+C}}}∫eaxdx=1aeax+C{\displaystyle {\rm {\int e^{ax}dx={\frac {1}{a}}e^{ax}+C}}}∫axdx=1lnaax+C{\displaystyle {\rm {\int a^{x}dx={\frac {1}{\ln a}}a^{x}+C}}}∫1xdx=lnx+C{\displaystyle {\rm {\int {\frac {1}{x}}dx=\ln x+C}}}∫1ax+bdx=1aln(ax+b)+C{\displaystyle {\rm {\int {\frac {1}{ax+b}}dx={\frac {1}{a}}\ln(ax+b)+C}}}∫sinxdx=−cosx+C{\displaystyle {\rm {\int \sin xdx=-\cos x+C}}}∫sinaxdx=−1acosax+C{\displaystyle {\rm {\int \sin axdx=-{\frac {1}{a}}\cos ax+C}}}∫cosxdx=sinx+C{\displaystyle {\rm {\int \cos xdx=\sin x+C}}}∫cosaxdx=1asinax+C{\displaystyle {\rm {\int \cos axdx={\frac {1}{a}}\sin ax+C}}}In=∫0π2sinnxdx=∫0π2cosnxdx=n−1nIn−2{\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}}}}∫dxcos2x=∫sec2xdx=tanx+C{\displaystyle {\rm {\int {\frac {dx}{\cos ^{2}x}}=\int \sec ^{2}xdx=\tan x+C}}}∫dxsin2x=∫csc2xdx=−cotx+C{\displaystyle {\rm {\int {\frac {dx}{\sin ^{2}x}}=\int \csc ^{2}xdx=-\cot x+C}}}∫secx⋅tanxdx=secx+C{\displaystyle {\rm {\int \sec x\cdot \tan xdx=\sec x+C}}}∫cscx⋅cotxdx=−cscx+C{\displaystyle {\rm {\int \csc x\cdot \cot xdx=-\csc x+C}}}∫tanxdx=−ln|cosx|+C{\displaystyle {\rm {\int \tan xdx=-\ln |\cos x|+C}}}∫cotxdx=ln|sinx|+C{\displaystyle {\rm {\int \cot xdx=\ln |\sin x|+C}}}∫secxdx=ln|sec(x)+tan(x)|+C{\displaystyle {\rm {\int \sec xdx=\ln |\sec(x)+\tan(x)|+C}}}∫cscxdx=ln|csc(x)−cot(x)|+C{\displaystyle {\rm {\int \csc xdx=\ln |\csc(x)-\cot(x)|+C}}}∫sinhxdx=coshx+C{\displaystyle {\rm {\int \sinh xdx=\cosh x+C}}}∫coshxdx=sinhx+C{\displaystyle {\rm {\int \cosh xdx=\sinh x+C}}}∫dx1−x2=arcsinx+C{\displaystyle {\rm {\int {\frac {dx}{\sqrt {1-x^{2}}}}=\arcsin x+C}}}∫dxa2−x2=arcsinxa+C{\displaystyle {\rm {\int {\frac {dx}{\sqrt {a^{2}-x^{2}}}}=\arcsin {\frac {x}{a}}+C}}}∫dx1+x2=arctanx+C{\displaystyle {\rm {\int {\frac {dx}{1+x^{2}}}=\arctan x+C}}}∫dxa2+x2=1aarctanxa+C{\displaystyle {\rm {\int {\frac {dx}{a^{2}+x^{2}}}={\frac {1}{a}}\arctan {\frac {x}{a}}+C}}}∫dxa2−x2=12alna+xa−x+C{\displaystyle {\rm {\int {\frac {dx}{a^{2}-x^{2}}}={\frac {1}{2a}}\ln {\frac {a+x}{a-x}}+C}}}∫dxx2−a2=12aln|x−ax+a|+C{\displaystyle {\rm {\int {\frac {dx}{x^{2}-a^{2}}}={\frac {1}{2a}}\ln |{\frac {x-a}{x+a}}|+C}}}∫dxx2±a2=ln(x+x2±a2)+C{\displaystyle {\rm {\int {\frac {dx}{\sqrt {x^{2}\pm a^{2}}}}=\ln(x+{\sqrt {x^{2}\pm a^{2}}})+C}}}∫x2+a2dx=x2x2+a2+a22ln(x+x2+a2)+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}}}∫x2−a2dx=x2x2−a2−a22ln|x+x2−a2|+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}}}∫a2−x2dx=x2a2−x2+a22arcsinxa+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}}}
