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![{\displaystyle \int c\cdot f(x)\mathrm {d} x=c\cdot \int f(x)\mathrm {d} x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Nyge4aDnDzDm4ztJCzjJDa2i4otCQaAzCzDe5oqw5oDmOztsQzqs0)
![{\displaystyle \int {\big (}f(x)\pm g(x){\big )}\mathrm {d} x=\int f(x)\mathrm {d} x\pm \int g(x)\mathrm {d} x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DajC5zNhDagdEnAnEngrAnthDnji5njsOoqa0zta1aNlEoNi4aDw2)
where ![{\displaystyle F'=f}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DatmPaNhEzNlFz2eQoDvDaNC2zDJAzta4zAePzga0zjBFztePz2s3)
![{\displaystyle \int u\,dv=uv-\int v\,du}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EngvEaDG3aqo0yjwQygw2ytBCothCytvFnDeQzNnFatsQajG5yjBF)
![{\displaystyle \int \mathrm {d} x=x+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84zjnBagiOoNlAnto4zAs3oNK5ajiOoDa4zNw1ajs2aqsPyqdAnAi3)
![{\displaystyle \int a\,\mathrm {d} x=ax+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83ajnByjnAoDzFz2vFzNeNytlBzDGNzgzBaNs1aga2yto4aDs3nDm5)
![{\displaystyle \int x^{n}\mathrm {d} x={\frac {x^{n+1}}{n+1}}+C\qquad ({\text{for }}n\neq -1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QatFBzgzCzjKQnDFAoNG0zjFCzga4ajsPngw0njKOygiQzDzCyts2)
![{\displaystyle \int {\frac {\mathrm {d} x}{x}}=\ln |x|+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QnDCOoDC2yjK2zDBAngs1yqs2ngoQzAo4oAvFaDa1zDC2zqe0njmP)
![{\displaystyle \int {\frac {\mathrm {d} x}{ax+b}}={\frac {\ln |ax+b|}{a}}+C\qquad ({\text{for }}a\neq 0)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CzgwQntaOyjrCzjG1njC5otePaDe0oDK5zqrEaNK3yqzAngsOoDhE)
![{\displaystyle \int \sin(x)\mathrm {d} x=-\cos(x)+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AoAeOzqeQnAw3oNaQoDG1z2eNatC1zgwPaDa0aqrEoAaOnqzCytGP)
![{\displaystyle \int \cos(x)\mathrm {d} x=\sin(x)+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82ytw0ygvAaDrDzDe5yga4nAo4o2nCyqo0oNw5yqhFyqaOzNBEzjiO)
![{\displaystyle \int \tan(x)\mathrm {d} x=\ln |\sec(x)|+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83zDa4ytzFa2s1oDCNnjaNajCOaDvEajzBoNs5oDdFntwOaNzDntG2)
![{\displaystyle \int \sin ^{2}(x)\mathrm {d} x=\int {\frac {1-\cos(2x)}{2}}\mathrm {d} x={\frac {x}{2}}-{\frac {\sin(2x)}{4}}+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EaDlEzgvEnjePagoNztlCoDdAajmNo2hFzNa1zjoQzNlAntG2oDGO)
![{\displaystyle \int \cos ^{2}(x)\mathrm {d} x=\int {\frac {1+\cos(2x)}{2}}\mathrm {d} x={\frac {x}{2}}+{\frac {\sin(2x)}{4}}+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81njGNzjC3zDm3z2zAztaPngiPajGOo2hBoDnAnDBBzqi2zjzBngwQ)
![{\displaystyle \int \tan ^{2}(x)\mathrm {d} x=\tan(x)-x+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Oyqe3ats0oDhDzgaOoNFBaNKOoDmPyto4zNs0yte3zDvAnja2ytGQ)
![{\displaystyle \int \sec(x)\mathrm {d} x=\ln {\Big |}\sec(x)+\tan(x){\Big |}+C=\ln \left|\tan \left({\frac {x}{2}}+{\frac {\pi }{4}}\right)\right|+C=2\mathrm {artanh} \left(\tan \left({\frac {x}{2}}\right)\right)+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84atrBaNlBnArFotGNa2dCaNrFaDo3nts5aje3zAo5ztK4ytKOnDw4)
![{\displaystyle \int \csc(x)\mathrm {d} x=-\ln {\Big |}\csc(x)+\cot(x){\Big |}+C=\ln \left|\tan \left({\frac {x}{2}}\right)\right|+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82aAzBntiNnjBEaNFAzqs1njKOaqaNnjFFa2dBygiQnAe1yqnDnjK3)
![{\displaystyle \int \cot(x)\mathrm {d} x=\ln |\sin(x)|+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CzDi2aqi2agzCzgvFaDFByqw3aNaNzqi1otm5ytmQzje2zjG2zgo0)
![{\displaystyle \int \sec ^{2}(ax)\mathrm {d} x={\frac {\tan(ax)}{a}}+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82zjeOaja5zti3nqhDnga4yqw5oDaQyjlBaArCagw4nDiPzNFCnDmO)
![{\displaystyle \int \csc ^{2}(ax)\mathrm {d} x=-{\frac {\cot(ax)}{a}}+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FytwQz2rCnjCPnAwQnAoQzjFDoAw5z2hBzNKNzjKOyjm1yjG3aNrA)
![{\displaystyle \int \cot ^{2}(ax)\mathrm {d} x=-x-{\frac {\cot(ax)}{a}}+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82njG2aje2yqePatK5nDsPzDi3otK3aNdCzDePajo3oDJAnga2z2vB)
![{\displaystyle \int \sec(x)\tan(x)\mathrm {d} x=\sec(x)+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85zjvCzDaNoAhCzNGNztrAa2s4ytmOygeOnjGOajG2o2s4zAo2ntJF)
![{\displaystyle \int \sec(x)\csc(x)\mathrm {d} x=\ln |\tan(x)|+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84aDeOzNm3aDC4aNG5aDFEotrFoDiQa2rCaAaQoto3otmOoAnBzqi0)
![{\displaystyle \int \sin ^{n}(x)\mathrm {d} x=-{\frac {\sin ^{n-1}(x)\cos(x)}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}(x)\mathrm {d} x+C\qquad ({\text{for }}n>0)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Aygw2ztvBngi5oDC4zgi1njnFzqvEotFDyqhFatJEzqi4zjeNzAsP)
![{\displaystyle \int \cos ^{n}(x)\mathrm {d} x=-{\frac {\cos ^{n-1}(x)\sin(x)}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}(x)\mathrm {d} x+C\qquad ({\text{for }}n>0)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80njiQngw1aqw3zjwNoDw0ngo1o2oNygvAotGQyje3ati2yts0yqaQ)
![{\displaystyle \int \tan ^{n}(x)\mathrm {d} x={\frac {\tan ^{n-1}(x)}{(n-1)}}-\int \tan ^{n-2}(x)\mathrm {d} x+C\qquad ({\text{for }}n\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FatJDaqa4ytm2zNK1nAa3oDoOaNKQoDzCzArFatm3aqrEots5oNGN)
![{\displaystyle \int \sec ^{n}(x)\mathrm {d} x={\frac {\sec ^{n-1}(x)\sin(x)}{n-1}}+{\frac {n-2}{n-1}}\int \sec ^{n-2}(x)\mathrm {d} x+C\qquad ({\text{for }}n\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83ytsPnDdByjmPotm0othEzgvFngi4aNi3nAwQage5zqo0atiOzNoN)
![{\displaystyle \int \csc ^{n}(x)\mathrm {d} x=-{\frac {\csc ^{n-1}(x)\cos(x)}{n-1}}+{\frac {n-2}{n-1}}\int \csc ^{n-2}(x)\mathrm {d} x+C\qquad ({\text{for }}n\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82zgo0nDm0yqwPagrAoAiQaNm5aAdDzDw1aje2njs1aga2otiNoArB)
![{\displaystyle \int \cot ^{n}(x)\mathrm {d} x=-{\frac {\cot ^{n-1}(x)}{n-1}}-\int \cot ^{n-2}(x)\mathrm {d} x+C\qquad ({\text{for }}n\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81oqsQygo1nAe2yqzFotrEaqo1oNa0nAnFaAzCngiPyjJFajdBaDw3)
![{\displaystyle a^{2}\int x^{n}\sin(ax)\mathrm {d} x=nx^{n-1}\sin(ax)-ax^{n}\cos(ax)-n(n-1)\int x^{n-2}\sin(ax)\mathrm {d} x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Pajm0aqiOoDGNngi4zDm2zjwNzqhFoNdCzjaQzgs5aNzFngzBztGP)
![{\displaystyle a^{2}\int x^{n}\cos(ax)\mathrm {d} x=ax^{n}\sin(ax)+nx^{n-1}\cos(ax)-n(n-1)\int x^{n-2}\cos(ax)\mathrm {d} x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NnqvCaqhFajaNzNFBz2oQo2e2nghEatvAzjhFaDBFoAiNztdFags5)
![{\displaystyle \int \sin ^{n}(x)\mathrm {d} x=-\cos(x)_{2}F_{1}\left({\frac {1}{2}},{\frac {1-n}{2}};{\frac {3}{2}};\cos ^{2}(x)\right)+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QoNnFoDGPoDzBytG5njdFaAa0nAzEoDvAntnFzjlCygw1ntrCyjo4)
![{\displaystyle \int \cos ^{n}(x)\mathrm {d} x=-{\frac {1}{n+1}}\mathrm {sgn} (\sin(x))\cos ^{n+1}(x)_{2}F_{1}\left({\frac {1}{2}},{\frac {n+1}{2}};{\frac {n+3}{2}};\cos ^{2}(x)\right)+C\qquad ({\text{for }}n\neq -1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81zDa4aqe2zta0yqsPngrFaqzDzArEaNwPzti0oNJFajFEzNGQoNmN)
![{\displaystyle \int \tan ^{n}(x)\mathrm {d} x={\frac {1}{n+1}}\tan ^{n+1}(x)_{2}F_{1}\left(1,{\frac {n+1}{2}};{\frac {n+3}{2}};-\tan ^{2}(x)\right)+C\qquad ({\text{for }}n\neq -1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PnArFnDC1atw5zqiNoDCOyje3zNFCzgrEnjaOytFDatBAygzEztlE)
![{\displaystyle \int \csc ^{n}(x)\mathrm {d} x=-\cos(x)_{2}F_{1}\left({\frac {1}{2}},{\frac {n+1}{2}};{\frac {3}{2}};\cos ^{2}(x)\right)+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DzgzAaNKQyjaOnAs4ygaPoDKNajwNyqs3oAeQoDG5aqoPnDvCnjK1)
![{\displaystyle \int \sec ^{n}(x)\mathrm {d} x=\sin(x)_{2}F_{1}\left({\frac {1}{2}},{\frac {n+1}{2}};{\frac {3}{2}};\sin ^{2}(x)\right)+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CnAoQyjnEaDaQyjGQzqw2zjzDnArCz2i4a2zFaqzFoqeNyje2zDw0)
![{\displaystyle \int \cot ^{n}(x)\mathrm {d} x=-{\frac {1}{n+1}}\cot ^{n+1}(x)_{2}F_{1}\left(1,{\frac {n+1}{2}};{\frac {n+3}{2}};-\cot ^{2}(x)\right)+C\qquad ({\text{for }}n\neq -1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80oqi2ngzBygi2njvByja1zgrFygvCo2eOyqeOz2a1zjw1aDiOzDe1)
Where
is the hypergeometric function and
is the sign function.
![{\displaystyle \int {\frac {\mathrm {d} x}{\sqrt {1-x^{2}}}}=\arcsin(x)+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FajFCzAvEotJBoqe1zDlEyjFEaNi4nDC5ajsPnDlEngwQajK2aNnD)
![{\displaystyle \int {\frac {\mathrm {d} x}{\sqrt {a^{2}-x^{2}}}}=\arcsin \left({\tfrac {x}{a}}\right)+C\qquad ({\text{for }}a\neq 0)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85otdEoDsOatm5oDGQnjwPaNwNaNBAaqiNyte0aDwNztvCoDaPzjsO)
![{\displaystyle \int {\frac {\mathrm {d} x}{1+x^{2}}}=\arctan(x)+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84zgnEzti5yqe5oAdEzgw5yta1yga0yqs0oqwQztw3nAaQzgePnDBB)
![{\displaystyle \int {\frac {\mathrm {d} x}{a^{2}+x^{2}}}={\frac {\arctan \left({\tfrac {x}{a}}\right)}{a}}+C\qquad ({\text{for }}a\neq 0)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Po2sNztdDzqs2oNw1aqzDzDG1yjeOajeOaDzDnDzAztC1ygnFajBA)
![{\displaystyle \int e^{x}\mathrm {d} x=e^{x}+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80nqeQatFAaqo1zgsOnDsPoNrBzNe4njnAygnCz2vBngrBnAa4aqnA)
![{\displaystyle \int e^{ax}\mathrm {d} x={\frac {e^{ax}}{a}}+C\qquad ({\text{for }}a\neq 0)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83aAoNoNm1ygvAyqnDytJCzji3a2e3njiQntiOngrBaArAzjsPzqe1)
![{\displaystyle \int a^{x}\mathrm {d} x={\frac {a^{x}}{\ln(a)}}+C\qquad ({\text{for }}a>0,a\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PngzAnqrFyqe0aNmNnDnEzthDaqwPaAsPntzBnDm2oDa1atePotdF)
![{\displaystyle \int \ln(x)\mathrm {d} x=x\ln(x)-x+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82zNG0yjzFnDdDzDlBzteQyjG1aja3o2s0oNnAaDCPoDiNzNe3aDw2)
![{\displaystyle \int e^{x}\sin(x)\mathrm {d} x={\frac {e^{x}}{2}}(\sin(x)-\cos(x))+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82oAsQnqdBoNw1nDnDataNoDa5o2aOaAi1ngiNztlEoAe1ygnEajwO)
![{\displaystyle \int e^{x}\cos(x)\mathrm {d} x={\frac {e^{x}}{2}}(\sin(x)+\cos(x))+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81aAi1zteOaNJDotmNzNe4yjsQzDw2oNFDythAotw0oDlDztsNz2eQ)
![{\displaystyle \int x^{n}e^{ax}\mathrm {d} x={\frac {1}{a}}x^{n}e^{ax}-{\frac {n}{a}}\int x^{n-1}e^{ax}\mathrm {d} x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PnDFEytvBntJEzNG2ztrDotdCaAw2zjrFzqs0oAaPnjaQntJAaNeP)
![{\displaystyle \int \arcsin(x)\mathrm {d} x=x\arcsin(x)+{\sqrt {1-x^{2}}}+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FoNe4zAnDaDGNaAdCzDiQajGQoAdBoti4zqePngdDzNi4aDvFyjo4)
![{\displaystyle \int \arccos(x)\mathrm {d} x=x\arccos(x)-{\sqrt {1-x^{2}}}+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Azqa5njC3zDw5aDi1ntC1zqnBnDe0nAe3yjGOzNoOytlCatG5yqnF)
![{\displaystyle \int \arctan(x)\mathrm {d} x=x\arctan(x)-{\frac {1}{2}}\ln |1+x^{2}|+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82atmOzAzBnDlBaAhBnqiPotJCaNa4aAvAoqvCoqwQnDvFyteQoDG1)
![{\displaystyle \int \operatorname {arccsc}(x)\mathrm {d} x=x\operatorname {arccsc}(x)+\ln \left|x+x{\sqrt {1-{\frac {1}{x^{2}}}}}\right|+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84a2hDntm2nDC2oNsOatvDz2w3ytK0ygs3oDi3zjm3njCQzgs4zNm2)
![{\displaystyle \int \operatorname {arcsec}(x)\mathrm {d} x=x\operatorname {arcsec}(x)-\ln \left|x+x{\sqrt {1-{\frac {1}{x^{2}}}}}\right|+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NoAa2atvEygwNaNoNaNwNnqs5ygeNzDJCzDmOajhEzte2otoOaqvF)
![{\displaystyle \int \operatorname {arccot}(x)\mathrm {d} x=x\operatorname {arccot}(x)+{\frac {1}{2}}\ln |1+x^{2}|+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CaNzEyjBCzAaOoDG2nje3nDhDnAaOoDe5ytC0zjo2nqnFaNeQz2e0)
![{\displaystyle \int \sinh(x)\mathrm {d} x=-i\int \sin(ix)\mathrm {d} x=\cos(ix)+C=\cosh(x)+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AnjmQaDa1njC1ygo2oqoQztnAoqw1aNFBoqdFaqsPztm3ztvBnjwN)
![{\displaystyle \int \cosh(x)\mathrm {d} x=\int \cos(ix)\mathrm {d} x=-i\sin(ix)+C=\sinh(x)+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80zNi0aDCQzqsNnDs2ytJBoqhCyge3oNFBztKOoqdAoqw4z2w1aqsP)
![{\displaystyle \int \tanh(x)\mathrm {d} x=-i\int \tan(ix)\mathrm {d} x=\log \left|\cos(ix)\right|+C=\log \left|\cosh(x)\right|+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PotrCnqa0z2vBnDvDnDnAa2hFaDG2z2w0aAi5ytnBzti4oqe5oqdA)
![{\displaystyle \int \mathrm {csch} (x)\mathrm {d} x=i\int \csc(ix)\mathrm {d} x=\log \left|-i\tan \left({\frac {ix}{2}}\right)\right|+C=\log \left|\tanh \left({\frac {x}{2}}\right)\right|+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83zNs2yqvDnteNathCnjoNzNdDaNCQyji5ytK4ztFDztGQnqoNatrE)
![{\displaystyle \int \mathrm {sech} (x)\mathrm {d} x=\int \sec(ix)\mathrm {d} x=2\mathrm {artanh} \left(-i\tan \left({\frac {x}{2}}i\right)\right)+C=2\arctan \left(\tanh \left({\frac {x}{2}}\right)\right)+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EzDK5ytlCnqsOoNm2yjC4agsPzNeNnAvAygo3oDa1yjvBzqa0z2dD)
![{\displaystyle \int \mathrm {coth} (x)\mathrm {d} x=i\int \cot(ix)\mathrm {d} x=\log \left|-i\sin(ix)\right|+C=\log \left|\sinh(x)\right|+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83othAagePoAwPoNlCa2s3ngoOngdBnDi4nAnFnqiPoqa3ntm5oDGP)
![{\displaystyle \int \mathrm {arsinh} (x)\mathrm {d} x=x\mathrm {arsinh} (x)-{\sqrt {x^{2}+1}}+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EoDi1aNG2ate1aNw2aNBEoDvBaNoPzDeQnghEota3aqnCyjFBataO)
![{\displaystyle \int \mathrm {arcosh} (x)\mathrm {d} x=x\mathrm {arcosh} (x)-{\sqrt {x^{2}-1}}+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83nDm5aAiNaDnCzjeQaDdCntlBaNrCnDnCa2sOaDs2zNo2zqwOngrB)
![{\displaystyle \int \mathrm {artanh} (x)\mathrm {d} x=x\mathrm {artanh} (x)+{\frac {1}{2}}\ln(1-x^{2})+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NoqaOaDa3yqaQyqzDngsNztlBagnBaDnDzAvBnqnDz2wNyjlAoNdC)
![{\displaystyle \int \mathrm {arcsch} (x)\mathrm {d} x=x\mathrm {arcsch} (x)+|\mathrm {arsinh} (x)|+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81nDa5z2e0a2s0oNo2aAi2nAzDoDaOzNvDz2vAzjhDaghFzAeNzDvD)
![{\displaystyle \int \mathrm {arsech} (x)\mathrm {d} x=x\mathrm {arsech} (x)+\arcsin(x)+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AnDGOoDrBaNC1ajwPnjrCatKPajGPytJEo2zEyqe4ztJCaNwQoqdE)
![{\displaystyle \int \mathrm {artanh} (x)\mathrm {d} x=x\mathrm {arcoth} (x)+{\frac {1}{2}}\ln(x^{2}-1)+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EoqvDoqvDaqe3yghDyjnEzNG0zDhBzNdEoDG3aAsPztKOzNzEoqzA)
, where
is the sign function.
, where
is the Riemann zeta function.
![{\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}\mathrm {d} x={\sqrt {\pi }}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DyqiPnteNo2i4nge3njK3oNeNoDw0zAi4aAw1zgnEoqo4agi2zghF)
, where
is the gamma function.
![{\displaystyle \int _{0}^{\infty }t^{s-1}e^{-t}\mathrm {d} t=\Gamma (s)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OytFAz2dAoDw2zDdBo2sQzDe4nDo5ata0aDnCzjK3nqsQzgo1zDhB)
, where
is the modified Bessel function of the first kind.
![{\displaystyle \int _{0}^{\infty }{\frac {\sin(x)}{x}}\mathrm {d} x={\frac {\pi }{2}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84nDG2njKNaDo4zAnCaDzCnAsOnjBAoDs4ygwQzga5ytzBzjm0o2zA)