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Questa pagina contiene una tavola di integrali indefiniti di funzioni trigonometriche.
- Per altri integrali vedi Integrale § Tavole di integrali.
In questa pagina si assume che
sia una costante diversa da 0.
![{\displaystyle \int \sin(cx)\;\mathrm {d} x=-{\frac {\cos(cx)}{c}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84zNdDa2hBate3nDBEz2sNyghBzNe0nAwQytlCoqvAoDC3agi2zNvD)
![{\displaystyle \int \sin ^{2}x\;\mathrm {d} x={\frac {1}{2}}(x-\sin x\cos x)+C}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83zjwQoqdCnga3njvEyqzAzje2zDm5yjJFotCQzDwQaDe2ztC0aqo3)
![{\displaystyle \int \sin ^{n}(cx)\;\mathrm {d} x=-{\frac {\sin ^{n-1}(cx)\cos(cx)}{nc}}+{\frac {n-1}{n}}\int \sin ^{n-2}(cx)\;\mathrm {d} x\qquad ({\text{per }}n>0)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84ytnAaqiPota5nDrFaNrEotrBatnEytC2yqwQygs4ngs0yqnBztaN)
![{\displaystyle \int x\sin(cx)\;\mathrm {d} x={\frac {\sin(cx)}{c^{2}}}-{\frac {x\cos(cx)}{c}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FyqvFnge5yqvCzgnBatvEzNi5yqi5a2i4ajvDzqi4aAw5zAdFaAe5)
![{\displaystyle \int x^{n}\sin(cx)\;\mathrm {d} x=-{\frac {x^{n}}{c}}\cos(cx)+{\frac {n}{c}}\int x^{n-1}\cos(cx)\;\mathrm {d} x\qquad ({\text{per }}n>0)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80ztaNaDBAytC2aqwOnDeQaNFEnDi3nDaNato2aDa3ygs0aNwQntK3)
![{\displaystyle \int {\frac {\sin(cx)}{x}}\mathrm {d} x=\sum _{i=0}^{\infty }(-1)^{i}{\frac {(cx)^{2i+1}}{(2i+1)\cdot (2i+1)!}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Nzga0aDwPnAiNoNw5ato2oqo1zAe3zjK1zNo5oAs0otdAo2w0yjGN)
![{\displaystyle \int {\frac {\sin(cx)}{x^{n}}}\mathrm {d} x=-{\frac {\sin cx}{(n-1)x^{n-1}}}+{\frac {c}{n-1}}\int {\frac {\cos(cx)}{x^{n-1}}}\mathrm {d} x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81o2eOajiOztlBa2o0otlCaga0zqsNnghDyjnEzAnDz2hEoNdCaDKP)
![{\displaystyle \int {\frac {\mathrm {d} x}{\sin(cx)}}={\frac {1}{c}}\ln \left|\tan {\frac {cx}{2}}\right|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83aAs3nDKQaAoNnqo3yjs4z2a0aDaQajmNytdCnAo5zDvCajnByjaO)
![{\displaystyle \int {\frac {\mathrm {d} x}{\sin ^{n}(cx)}}={\frac {\cos(cx)}{c(1-n)\sin ^{n-1}(cx)}}+{\frac {n-2}{n-1}}\int {\frac {\mathrm {d} x}{\sin ^{n-2}cx}}\qquad ({\text{per }}n>1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82oDm5nAzCzqi0ygrCo2hBntG1nDvDnja2yjKPzgzBoNK2yqe4zjs4)
![{\displaystyle \int {\frac {\mathrm {d} x}{1\pm \sin(cx)}}={\frac {1}{c}}\tan \left({\frac {cx}{2}}\mp {\frac {\pi }{4}}\right)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Pz2hBatm0ats3o2eOoAoNzNaPoAzFnAiQzjzAzjCPytvCzAzDzDiQ)
![{\displaystyle \int {\frac {x\;\mathrm {d} x}{1+\sin(cx)}}={\frac {x}{c}}\tan \left({\frac {cx}{2}}-{\frac {\pi }{4}}\right)+{\frac {2}{c^{2}}}\ln \left|\cos \left({\frac {cx}{2}}-{\frac {\pi }{4}}\right)\right|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EnjvFztCOztFDoDw4oDwQztBBoti5z2dAnDG3ztiOagaOnAs3aAhD)
![{\displaystyle \int {\frac {x\;\mathrm {d} x}{1-\sin cx}}={\frac {x}{c}}\cot \left({\frac {\pi }{4}}-{\frac {cx}{2}}\right)+{\frac {2}{c^{2}}}\ln \left|\sin \left({\frac {\pi }{4}}-{\frac {cx}{2}}\right)\right|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QzDnDnqvBnqw1nDhCote4ytsOyjvEzqePzqa3aAdDzjrEoNw1z2s4)
![{\displaystyle \int {\frac {\sin cx\;\mathrm {d} x}{1\pm \sin cx}}=\pm x+{\frac {1}{c}}\tan \left({\frac {\pi }{4}}\mp {\frac {cx}{2}}\right)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NzjrFajiQzji4yjiQa2nBajBCytFDzqzAngsNaDCOngaQytnEnAaP)
![{\displaystyle \int \sin c_{1}x\sin c_{2}x\;\mathrm {d} x={\frac {\sin(c_{1}-c_{2})x}{2(c_{1}-c_{2})}}-{\frac {\sin(c_{1}+c_{2})x}{2(c_{1}+c_{2})}}\qquad ({\text{per }}|c_{1}|\neq |c_{2}|)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Azga5zjGPajzFzgnAytK5nDsPztrCoArFoNvDytG1nte3oDK5njrE)
Lo stesso argomento in dettaglio: Coseno.
![{\displaystyle \int \cos(cx)\;\mathrm {d} x={\frac {\sin(cx)}{c}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84njhEotJDoAa5z2s2yjmOa2i0aNeQaDnFyga5zjw0atw5ztlDzjo5)
![{\displaystyle \int \cos ^{n}(cx)\;\mathrm {d} x={\frac {\cos ^{n-1}(cx)\sin(cx)}{nc}}+{\frac {n-1}{n}}\int \cos ^{n-2}(cx)\;\mathrm {d} x\qquad ({\text{per }}n>0)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EytJCyte0ntGQoqsPzAa2ygw0yjmQnjiQotnFoAw4njs1yjrDzDo1)
![{\displaystyle \int x\cos(cx)\;\mathrm {d} x={\frac {\cos(cx)}{c^{2}}}+{\frac {x\sin(cx)}{c}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PyjrEnDi4zjGNnDo5oqo2a2vAytsNoNzAoDiOygzEajK0nAeNaDmN)
![{\displaystyle \int x^{n}\cos(cx)\;\mathrm {d} x={\frac {x^{n}\sin(cx)}{c}}-{\frac {n}{c}}\int x^{n-1}\sin(cx)\;\mathrm {d} x}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81nja0nga5zti0zDJAytwQatK3nDBEyjw1zjw4aNBCaDw3njw2ajK5)
![{\displaystyle \int {\frac {\cos(cx)}{x}}\mathrm {d} x=\ln |cx|+\sum _{i=1}^{\infty }(-1)^{i}{\frac {(cx)^{2i}}{2i\cdot (2i)!}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Aago0zgsNzNs5aqw4zgoQa2sNotK5nDo4njrCz2ePoAo1zDoNzDs3)
![{\displaystyle \int {\frac {\cos(cx)}{x^{n}}}\mathrm {d} x=-{\frac {\cos(cx)}{(n-1)x^{n-1}}}-{\frac {c}{n-1}}\int {\frac {\sin(cx)}{x^{n-1}}}\mathrm {d} x\qquad ({\text{per }}n\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EaAsNoAsOnqw5z2s0z2w2yqsNaDzAoDnBzgaQajBBnjK0yjFCzjs2)
![{\displaystyle \int {\frac {\mathrm {d} x}{\cos(cx)}}={\frac {1}{c}}\ln \left|\tan \left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)\right|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FztvEzAa5yqvFnto1oDGQz2o0aNC5zNrEyga3nqdCoDzEa2a2nAvB)
![{\displaystyle \int {\frac {\mathrm {d} x}{\cos ^{n}(cx)}}={\frac {\sin(cx)}{c(n-1)\cos ^{n-1}(cx)}}+{\frac {n-2}{n-1}}\int {\frac {\mathrm {d} x}{\cos ^{n-2}(cx)}}\qquad ({\text{per }}n>1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CzNJEoqaOoNwQoNC5yjzAota0otrEoqePoAvDo2a2oAnBzjJCztoP)
![{\displaystyle \int {\frac {\mathrm {d} x}{1+\cos(cx)}}={\frac {1}{c}}\tan {\frac {cx}{2}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QzNBDyqwOaDCOyqw5oDwNz2nAaDzBnjzEa2vBzgnBaje5oDdFaDFD)
![{\displaystyle \int {\frac {\mathrm {d} x}{1-\cos(cx)}}=-{\frac {1}{c}}\cot {\frac {cx}{2}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81zNoNyji2zNs0yjrFajlBzNs1aDaPntlBagzFnDiPoDe4ntm1oDs2)
![{\displaystyle \int {\frac {x\;\mathrm {d} x}{1+\cos(cx)}}={\frac {x}{c}}\tan({cx}/{2})+{\frac {2}{c^{2}}}\ln \left|\cos {\frac {cx}{2}}\right|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EyjlCnAiPnDBDzto1aqw4ygs2otw3yghDztiQytG2zgoQnji0otFF)
![{\displaystyle \int {\frac {x\;\mathrm {d} x}{1-\cos(cx)}}=-{\frac {x}{x}}\cot({cx}/{2})+{\frac {2}{c^{2}}}\ln \left|\sin {\frac {cx}{2}}\right|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83zqdAataNaDCQyts3zgs3ajmPoNzFzgwQaAdAagvAoDmQygzCajnB)
![{\displaystyle \int {\frac {\cos cx\;\mathrm {d} x}{1+\cos(cx)}}=x-{\frac {1}{c}}\tan {\frac {cx}{2}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83oAw0ajeOnDa2atiPygrEztlCaNrCyqvCzjBDoNaPoqvEntwQzto2)
![{\displaystyle \int {\frac {\cos cx\;\mathrm {d} x}{1-\cos(cx)}}=-x-{\frac {1}{c}}\cot {\frac {cx}{2}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PzDJEyqrCygw1yqaOntnCnDzEotdAzNm1a2hDztFBoAaQntJEytw0)
![{\displaystyle \int \cos c_{1}x\cos c_{2}x\;\mathrm {d} x={\frac {\sin(c_{1}-c_{2})x}{2(c_{1}-c_{2})}}+{\frac {\sin(c_{1}+c_{2})x}{2(c_{1}+c_{2})}}\qquad ({\text{per }}|c_{1}|\neq |c_{2}|)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Fyjo1zDzFyjG4a2w0njdCzqhDnjK1oqe0oNdBzNG0ngw3ytC2ntlF)
![{\displaystyle \int \tan cx\;\mathrm {d} x=-{\frac {1}{c}}\ln |\cos cx|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO83ygvAoAnAnts2yjlBygvAzNo5zgw4yqs3zts5otJAnjaNajm2ajw1)
![{\displaystyle \int \tan ^{n}cx\;\mathrm {d} x={\frac {1}{c(n-1)}}\tan ^{n-1}cx-\int \tan ^{n-2}cx\;\mathrm {d} x\qquad ({\text{per }}n\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82z2s5a2w0aDs5zjzFnje3ztBFoDaPoDw4yjnBotnDnAa3aNvFaNwQ)
![{\displaystyle \int {\frac {\mathrm {d} x}{\tan cx+1}}={\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx+\cos cx|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8OzgzCzjzCzqa4aDJDnDaQzgo2aNiQoDG2a2a0nqs5zje2zDm5a2dF)
![{\displaystyle \int {\frac {\mathrm {d} x}{\tan cx-1}}=-{\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx-\cos cx|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80yjrFoNnEytw1zji1zNaNyqePnAiNnAzDztdDoDlDaAa4oqa1ajJA)
![{\displaystyle \int {\frac {\tan cx\;\mathrm {d} x}{\tan cx+1}}={\frac {x}{2}}-{\frac {1}{2c}}\ln |\sin cx+\cos cx|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80zAhAz2iPnjC1zto5yta2ntC2oNC4agsQoDoPatdDnto1aDaQoNi0)
![{\displaystyle \int {\frac {\tan cx\;\mathrm {d} x}{\tan cx-1}}={\frac {x}{2}}+{\frac {1}{2c}}\ln |\sin cx-\cos cx|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Baqs5z2o4zDa3zjdAoNGQajo2ygiQnDdAntFEytiPaji0a2eOnjaP)
![{\displaystyle \int \sec {cx}\,\mathrm {d} x={\frac {1}{c}}\ln {\left|\sec {cx}+\tan {cx}\right|}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Fota0oNzEaqa1oNzDzjBCatnEatJFygePaDeOzja0ngdBoqw5oDlB)
![{\displaystyle \int \sec ^{n}{cx}\,\mathrm {d} x={\frac {\sec ^{n-1}{cx}\sin {cx}}{c(n-1)}}+{\frac {n-2}{n-1}}\int \sec ^{n-2}{cx}\,\mathrm {d} x\qquad {\text{per }}n\neq 1,c\neq 0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Pyje3yja2agrFyjKNaDzDaDs4ajGQatJAzqs1a2nAygdBzNi0atsQ)
![{\displaystyle \int \csc {cx}\,\mathrm {d} x=-{\frac {1}{c}}\ln {\left|\csc {cx}+\cot {cx}\right|}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FyjvCzDhDoqoQaNi4z2vBoNe0yjaOnqsQzAwOzqnCajaNota5aDGO)
![{\displaystyle \int \csc ^{n}{cx}\,\mathrm {d} x=-{\frac {\csc ^{n-1}{cx}\cos {cx}}{c(n-1)}}+{\frac {n-2}{n-1}}\int \csc ^{n-2}{cx}\,\mathrm {d} x\qquad {\text{per }}n\neq 1,c\neq 0}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NoNm0aDsOnqdDoDmNoNhFzts1zNG1nqi1ata2aDJBoNwNaAvCzDm3)
![{\displaystyle \int \cot cx\;\mathrm {d} x={\frac {1}{c}}\ln |\sin cx|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NnjnDzNlCz2s4njmQaqwNaAoOaqvFotC1z2iNnqo0zDhFatCQoNhA)
![{\displaystyle \int \cot ^{n}cx\;\mathrm {d} x=-{\frac {1}{c(n-1)}}\cot ^{n-1}cx-\int \cot ^{n-2}cx\;\mathrm {d} x\qquad ({\text{per }}n\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NoAi4zAdFyts4atm4zNFAaNaNz2nFnqhEyqwOajnDoDvFots5ztdE)
![{\displaystyle \int {\frac {\mathrm {d} x}{1+\cot cx}}=\int {\frac {\tan cx\;\mathrm {d} x}{\tan cx+1}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84ytmOajoOo2hFntvBnAs4zNCPote4zNzCagi1ztC5z2hAzNlDzNwP)
![{\displaystyle \int {\frac {\mathrm {d} x}{1-\cot cx}}=\int {\frac {\tan cx\;\mathrm {d} x}{\tan cx-1}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9EnDeQo2w5yjsNngs4yjdFaDBDajCQoqs1ntiPoNG0aDaNaDsNaArF)
![{\displaystyle \int {\frac {\mathrm {d} x}{\cos cx\pm \sin cx}}={\frac {1}{c{\sqrt {2}}}}\ln \left|\tan \left({\frac {cx}{2}}\pm {\frac {\pi }{8}}\right)\right|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Eyqs5oAe1yqa3a2dBatnBzNo0yqnDzNe5ygnBoNBAotCQoNnDyqe0)
![{\displaystyle \int {\frac {\mathrm {d} x}{(\cos cx\pm \sin cx)^{2}}}={\frac {1}{2c}}\tan \left(cx\mp {\frac {\pi }{4}}\right)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82z2s1ythEoNnEyjFBztBCa2oOajeNa2rEoNvFythByjrEytBCygnB)
![{\displaystyle \int {\frac {\cos cx\;\mathrm {d} x}{\cos cx+\sin cx}}={\frac {x}{2}}+{\frac {1}{2c}}\ln \left|\sin cx+\cos cx\right|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PnjG4nteNzDe4zgo5oqi5yjdFytvFnjwNaDBBzjJFyqo5otePotlC)
![{\displaystyle \int {\frac {\cos cx\;\mathrm {d} x}{\cos cx-\sin cx}}={\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\sin cx-\cos cx\right|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Pati5aAePzNa0aqhAnDe1aDe4ytC1oAiNnjs2zAa3yjw1aDw1nAnA)
![{\displaystyle \int {\frac {\sin cx\;\mathrm {d} x}{\cos cx+\sin cx}}={\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\sin cx+\cos cx\right|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80yte5njwNntm1aNnCytrBagdDzDFAo2w5oDC1aDG3nDaQo2e2o2aN)
![{\displaystyle \int {\frac {\sin cx\;\mathrm {d} x}{\cos cx-\sin cx}}=-{\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\sin cx-\cos cx\right|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81ajBEzDnBztdBz2wPaAiOzjnCaDJAzNw3aDlDnDw5nqhCoNo3zqhB)
![{\displaystyle \int {\frac {\cos cx\;\mathrm {d} x}{\sin cx(1+\cos cx)}}=-{\frac {1}{4c}}\tan ^{2}{\frac {cx}{2}}+{\frac {1}{2c}}\ln \left|\tan {\frac {cx}{2}}\right|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AztJAaqdCoAhFnArCnta0otvEnqvCzDs5nthDnDm2oqdDyjaNzNhD)
![{\displaystyle \int {\frac {\cos cx\;\mathrm {d} x}{\sin cx(1-\cos cx)}}=-{\frac {1}{4c}}\cot ^{2}{\frac {cx}{2}}-{\frac {1}{2c}}\ln \left|\tan {\frac {cx}{2}}\right|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8PztJByjKPaAi2atvCajKOnghDotJFaqvAaqeNnDG0zNzFytoQztsN)
![{\displaystyle \int {\frac {\sin cx\;\mathrm {d} x}{\cos cx(1+\sin cx)}}={\frac {1}{4c}}\cot ^{2}\left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)+{\frac {1}{2c}}\ln \left|\tan \left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)\right|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8NnDBBoqa1o2nEoqe2yjw0aghDaAa2aNaQoNlFaNFCnAo3ytnAotrA)
![{\displaystyle \int {\frac {\sin cx\;\mathrm {d} x}{\cos cx(1-\sin cx)}}={\frac {1}{4c}}\tan ^{2}\left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)-{\frac {1}{2c}}\ln \left|\tan \left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)\right|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80atrDytsOzqe2yje3njmOygiOztJEoDJAatmNzje1zDs1aqe5zqvA)
![{\displaystyle \int \sin cx\cos cx\;\mathrm {d} x={\frac {-1}{2c}}\cos ^{2}cx}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82zDwNnti5aNm0njJFaDe2zNK0ajG4ajoPaqo3ntwPygwQoqnFzDBC)
![{\displaystyle \int \sin c_{1}x\cos c_{2}x\;\mathrm {d} x=-{\frac {\cos(c_{1}+c_{2})x}{2(c_{1}+c_{2})}}-{\frac {\cos(c_{1}-c_{2})x}{2(c_{1}-c_{2})}}\qquad ({\text{per }}|c_{1}|\neq |c_{2}|)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9FagoOoNCOoqeOzDw0zAhCoDhAzNhFo2rEygsNoqdDntnDzjw3aDm0)
![{\displaystyle \int \sin ^{n}cx\cos cx\;\mathrm {d} x={\frac {1}{c(n+1)}}\sin ^{n+1}cx\qquad ({\text{per }}n\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BoAa5otBCztGNajs1aDs2nDzDatrFnjs4zDhFytzBaAeOyge2otCQ)
![{\displaystyle \int \sin cx\cos ^{n}cx\;\mathrm {d} x=-{\frac {1}{c(n+1)}}\cos ^{n+1}cx\qquad ({\text{per }}n\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80aNvBzDCOotnEyjGOnqw4oNm1zNrFz2e1oAe2a2vBoNm2ajvDa2a0)
![{\displaystyle \int \sin ^{n}cx\cos ^{m}cx\;\mathrm {d} x=-{\frac {\sin ^{n-1}cx\cos ^{m+1}cx}{c(n+m)}}+{\frac {n-1}{n+m}}\int \sin ^{n-2}cx\cos ^{m}cx\;\mathrm {d} x\qquad ({\text{per }}m,n>0)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Onjo3agaOyjG4athAaqa0zqrDntC4agzDaDhDnqrBnAvAo2wPaAaN)
- anche:
![{\displaystyle \int \sin ^{n}cx\cos ^{m}cx\;\mathrm {d} x={\frac {\sin ^{n+1}cx\cos ^{m-1}cx}{c(n+m)}}+{\frac {m-1}{n+m}}\int \sin ^{n}cx\cos ^{m-2}cx\;\mathrm {d} x\qquad ({\text{per }}m,n>0)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9DajwPo2sOoDGNyjCQatlBa2w2yqaPztFCnjmQyqwOaDa2aNG1ajo0)
![{\displaystyle \int {\frac {\mathrm {d} x}{\sin cx\cos cx}}={\frac {1}{c}}\ln \left|\tan cx\right|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84nAiQztvAajC4otG0yqwQzNw5zNdCajFDnje3oDi3yto1ngi5oti4)
![{\displaystyle \int {\frac {\mathrm {d} x}{\sin cx\cos ^{n}cx}}={\frac {1}{c(n-1)\cos ^{n-1}cx}}+\int {\frac {\mathrm {d} x}{\sin cx\cos ^{n-2}cx}}\qquad ({\text{per }}n\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO80aNnFaqhByqs0ajrAzge1yts2nAi4zDeQzgrEzqrAoDJBzAi4aqiO)
![{\displaystyle \int {\frac {\mathrm {d} x}{\sin ^{n}cx\cos cx}}=-{\frac {1}{c(n-1)\sin ^{n-1}cx}}+\int {\frac {\mathrm {d} x}{\sin ^{n-2}cx\cos cx}}\qquad ({\text{per }}n\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Ozts0oDwQztePaDm0nDK2nqi3oAvFzNvCaAdDaAi0zjzDzgsPzDG2)
![{\displaystyle \int {\frac {\sin cx\;\mathrm {d} x}{\cos ^{n}cx}}={\frac {1}{c(n-1)\cos ^{n-1}cx}}\qquad ({\text{per }}n\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9BaDs2ytK5oNnDaAwNnjvBnDi1zDw1yjdFntBAa2a1yjs5zjJEnjGN)
![{\displaystyle \int {\frac {\sin ^{2}cx\;\mathrm {d} x}{\cos cx}}=-{\frac {1}{c}}\sin cx+{\frac {1}{c}}\ln \left|\tan \left({\frac {\pi }{4}}+{\frac {cx}{2}}\right)\right|}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AnqdBntK2aqvDyqeQaNdDotG5yjw2oAdFataOyjm5atnEoNsQyjwN)
![{\displaystyle \int {\frac {\sin ^{2}cx\;\mathrm {d} x}{\cos ^{n}cx}}={\frac {\sin cx}{c(n-1)\cos ^{n-1}cx}}-{\frac {1}{n-1}}\int {\frac {\mathrm {d} x}{\cos ^{n-2}cx}}\qquad ({\text{per }}n\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9CnjCOa2eNatw0aqe2ntFCati3oqrByta0zjdAztJDaNe2ajhBnjGO)
![{\displaystyle \int {\frac {\sin ^{n}cx\;\mathrm {d} x}{\cos cx}}=-{\frac {\sin ^{n-1}cx}{c(n-1)}}+\int {\frac {\sin ^{n-2}cx\;\mathrm {d} x}{\cos cx}}\qquad ({\text{per }}n\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84otK2zDrFyjaOz2hCotC4oNvBaNhCzDo0ngrFoDFDzDGOags3zAeO)
![{\displaystyle \int {\frac {\sin ^{n}cx\;\mathrm {d} x}{\cos ^{m}cx}}={\frac {\sin ^{n+1}cx}{c(m-1)\cos ^{m-1}cx}}-{\frac {n-m+2}{m-1}}\int {\frac {\sin ^{n}cx\;\mathrm {d} x}{\cos ^{m-2}cx}}\qquad ({\text{per }}m\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Byje0aDsNyqiOzjFCzjlFoNCOagdBo2vCzAzCnjrDzqeQzNdBngnB)
- anche:
![{\displaystyle \int {\frac {\sin ^{n}cx\;\mathrm {d} x}{\cos ^{m}cx}}=-{\frac {\sin ^{n-1}cx}{c(n-m)\cos ^{m-1}cx}}+{\frac {n-1}{n-m}}\int {\frac {\sin ^{n-2}cx\;\mathrm {d} x}{\cos ^{m}cx}}\qquad {\text{per }}m\neq n)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81yte2zDFFnDm5zjKNats1zDmPztoPntlBygo2a2s2zDBDyjlDa2rF)
- anche:
![{\displaystyle \int {\frac {\sin ^{n}cx\;\mathrm {d} x}{\cos ^{m}cx}}={\frac {\sin ^{n-1}cx}{c(m-1)\cos ^{m-1}cx}}-{\frac {n-1}{m-1}}\int {\frac {\sin ^{n-1}cx\;\mathrm {d} x}{\cos ^{m-2}cx}}\qquad ({\text{per }}m\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Czgw3ytzFaDwQoqoNatoQaDKOzjaNnjvCaji4oAoOoDrAzDiOz2oO)
![{\displaystyle \int {\frac {\cos cx\;\mathrm {d} x}{\sin ^{n}cx}}=-{\frac {1}{c(n-1)\sin ^{n-1}cx}}\qquad ({\text{per }}n\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO85oNi4nDK5nDs0zqdFaNm2ntaQnjm5zgw1aNo2nqnBzDa0z2a4oNC0)
![{\displaystyle \int {\frac {\cos ^{2}cx\;\mathrm {d} x}{\sin cx}}={\frac {1}{c}}\left(\cos cx+\ln \left|\tan {\frac {cx}{2}}\right|\right)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Cota4ygi5zAa2a2oOzDBDaAwOnta3a2oNnqoPzNeNaDK3njvFzjC4)
![{\displaystyle \int {\frac {\cos ^{2}cx\;\mathrm {d} x}{\sin ^{n}cx}}=-{\frac {1}{n-1}}\left({\frac {\cos cx}{\sin ^{n-1}cx}}+\int {\frac {\mathrm {d} x}{\sin ^{n-2}cx}}\right)\qquad ({\text{per }}n\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84ztw2ntlDnjm2oqeQoNFBatC3zAo1zjaOaAaQoDiNa2s0yjnCnqaO)
![{\displaystyle \int {\frac {\cos ^{n}cx\;\mathrm {d} x}{\sin ^{m}cx}}=-{\frac {\cos ^{n+1}cx}{c(m-1)\sin ^{m-1}cx}}-{\frac {n-m-2}{m-1}}\int {\frac {\cos ^{n}cx\;\mathrm {d} x}{\sin ^{m-2}cx}}\qquad ({\text{per }}m\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Oo2rFzNBCotC3otdAzjdCzqhCoDi2zgdCatrFa2dEyjs2atJCztC3)
- anche:
![{\displaystyle \int {\frac {\cos ^{n}cx\;\mathrm {d} x}{\sin ^{m}cx}}={\frac {\cos ^{n-1}cx}{c(n-m)\sin ^{m-1}cx}}+{\frac {n-1}{n-m}}\int {\frac {\cos ^{n-2}cx\;\mathrm {d} x}{\sin ^{m}cx}}\qquad ({\text{per }}m\neq n)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO82nDJFaAe4ygvDatlFnDa5aqdBatCOnqvCntw4oDvFatG4ajvBoDG5)
- anche:
![{\displaystyle \int {\frac {\cos ^{n}cx\;\mathrm {d} x}{\sin ^{m}cx}}=-{\frac {\cos ^{n-1}cx}{c(m-1)\sin ^{m-1}cx}}-{\frac {n-1}{m-1}}\int {\frac {\cos ^{n-2}cx\;\mathrm {d} x}{\sin ^{m-2}cx}}\qquad ({\text{per }}m\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8QzNaOntwNoAw5otFFnjrBnqaNoNm2o2dDzDi4oNsOo2nBnja2zNFE)
![{\displaystyle \int \sin(cx)\tan(cx)\;\mathrm {d} x={\frac {\ln |\sec(cx)+\tan(cx)|-\sin(cx)}{c}}}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AaDsPyjm3nqw5ntiNoNwNaAa5oDhFotnEntm3oNePzqrBaNBCzNrC)
![{\displaystyle \int {\frac {\tan ^{n}(cx)}{\sin ^{2}(cx)}}\;\mathrm {d} x={\frac {\tan ^{n-1}(cx)}{c(n-1)}}\qquad ({\text{per }}n\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9AaDFEatG5njePaAdDytw0ztdCzNJAyjm4nDs5zDmQztdBytG5yga3)
![{\displaystyle \int {\frac {\tan ^{n}(cx)}{\cos ^{2}(cx)}}\;\mathrm {d} x={\frac {\tan ^{n+1}(cx)}{c(n+1)}}\qquad ({\text{per }}n\neq -1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO8Qoqi5agdFntG4zNG4ytKQzja4otoNoDi0njeOajo5otlBo2o1nqvF)
![{\displaystyle \int {\frac {\cot ^{n}(cx)}{\sin ^{2}(cx)}}\;\mathrm {d} x=-{\frac {\cot ^{n+1}(cx)}{c(n+1)}}\qquad ({\text{per }}n\neq -1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO9Anqa2zgi0aDFBatKNatmOotoNzjiQaji0agw3o2aPytvDoAa1oAwP)
![{\displaystyle \int {\frac {\cot ^{n}(cx)}{\cos ^{2}(cx)}}\;\mathrm {d} x={\frac {\tan ^{1-n}(cx)}{c(1-n)}}\qquad ({\text{per }}n\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO81aqdBzDBAyta4yjaNytvDoAs2oAa4zAdDaNiPaNaQaAsNaqzFaAi1)
![{\displaystyle \int {\frac {\tan ^{m}(cx)}{\cot ^{n}(cx)}}\;\mathrm {d} x={\frac {\tan ^{m+n-1}(cx)}{c(m+n-1)}}-\int {\frac {\tan ^{m-2}(cx)}{\cot ^{n}(cx)}}\;\mathrm {d} x\qquad ({\text{per }}m+n\neq 1)}](https://amansaja.com/index.php?q=Mfv0Kfa6bO93MqTXLqrCMqiSL3dZb2hQMu9Onpz0p3oPb21BngBFb21FJgGRKArSngrOb3z2nO84oNG0yjaNoNlAytBDoAeNyjGOatrDo2hDyqvDzNBCntK1zAeNz2sO)
- Murray R. Spiegel, Manuale di matematica, Etas Libri, 1974, pp. 75-82.