Início/Cálculo II/Apêndice A: tabela extensa de antiderivadas/A.2. Integrais trigonométricasConteúdoA.2. Integrais trigonométricasA.2.1 Potências pares (n=2n = 2n=2)Copiar LaTeX∫sen2u du=12 u−14 sen2u+C\int \operatorname{sen}^2 u\,du = \frac{1}{2}\,u - \frac{1}{4}\,\operatorname{sen} 2u + C∫sen2udu=21u−41sen2u+CVer detalhe →FórmulaCopiar LaTeX∫cos2u du=12 u+14 sen2u+C\int \cos^2 u\,du = \frac{1}{2}\,u + \frac{1}{4}\,\operatorname{sen} 2u + C∫cos2udu=21u+41sen2u+CVer detalhe →FórmulaCopiar LaTeX∫tan2u du=tanu−u+C\int \tan^2 u\,du = \tan u - u + C∫tan2udu=tanu−u+CVer detalhe →FórmulaCopiar LaTeX∫cot2u du=−cotu−u+C\int \cot^2 u\,du = -\cot u - u + C∫cot2udu=−cotu−u+CVer detalhe →A.2.2 Potências ímpares (n=3n = 3n=3)Copiar LaTeX∫sen3u du=−13(2+sen2u)cosu+C\int \operatorname{sen}^3 u\,du = -\frac{1}{3}\left(2 + \operatorname{sen}^2 u\right)\cos u + C∫sen3udu=−31(2+sen2u)cosu+CVer detalhe →FórmulaCopiar LaTeX∫cos3u du=13(2+cos2u)sen u+C\int \cos^3 u\,du = \frac{1}{3}\left(2 + \cos^2 u\right)\operatorname{sen}\,u + C∫cos3udu=31(2+cos2u)senu+CVer detalhe →FórmulaCopiar LaTeX∫tan3u du=12 tan2u+ln∣cosu∣+C\int \tan^3 u\,du = \frac{1}{2}\,\tan^2 u + \ln|\cos u| + C∫tan3udu=21tan2u+ln∣cosu∣+CVer detalhe →FórmulaCopiar LaTeX∫cot3u du=−12 cot2u−ln∣sen u∣+C\int \cot^3 u\,du = -\frac{1}{2}\,\cot^2 u - \ln|\operatorname{sen}\,u| + C∫cot3udu=−21cot2u−ln∣senu∣+CVer detalhe →FórmulaCopiar LaTeX∫sec3u du=12 secu tanu+12 ln∣secu+tanu∣+C\int \sec^3 u\,du = \frac{1}{2}\,\sec u\,\tan u + \frac{1}{2}\,\ln|\sec u + \tan u| + C∫sec3udu=21secutanu+21ln∣secu+tanu∣+CVer detalhe →FórmulaCopiar LaTeX∫csc3u du=−12 cscu cotu+12 ln∣cscu−cotu∣+C\int \csc^3 u\,du = -\frac{1}{2}\,\csc u\,\cot u + \frac{1}{2}\,\ln|\csc u - \cot u| + C∫csc3udu=−21cscucotu+21ln∣cscu−cotu∣+CVer detalhe →A.2.3 Fórmulas de reduçãoCopiar LaTeX∫sennu du=−1n senn−1u cosu+n−1n∫senn−2u du\int \operatorname{sen}^n u\,du = -\frac{1}{n}\,\operatorname{sen}^{n-1} u\,\cos u + \frac{n-1}{n}\int \operatorname{sen}^{n-2} u\,du∫sennudu=−n1senn−1ucosu+nn−1∫senn−2uduVer detalhe →FórmulaCopiar LaTeX∫cosnu du=1n cosn−1u sen u+n−1n∫cosn−2u du\int \cos^n u\,du = \frac{1}{n}\,\cos^{n-1} u\,\operatorname{sen}\,u + \frac{n-1}{n}\int \cos^{n-2} u\,du∫cosnudu=n1cosn−1usenu+nn−1∫cosn−2uduVer detalhe →FórmulaCopiar LaTeX∫tannu du=1n−1 tann−1u−∫tann−2u du\int \tan^n u\,du = \frac{1}{n-1}\,\tan^{n-1} u - \int \tan^{n-2} u\,du∫tannudu=n−11tann−1u−∫tann−2uduVer detalhe →
A.2.1 Potências pares (n=2n = 2n=2)Copiar LaTeX∫sen2u du=12 u−14 sen2u+C\int \operatorname{sen}^2 u\,du = \frac{1}{2}\,u - \frac{1}{4}\,\operatorname{sen} 2u + C∫sen2udu=21u−41sen2u+CVer detalhe →
FórmulaCopiar LaTeX∫cos2u du=12 u+14 sen2u+C\int \cos^2 u\,du = \frac{1}{2}\,u + \frac{1}{4}\,\operatorname{sen} 2u + C∫cos2udu=21u+41sen2u+CVer detalhe →
FórmulaCopiar LaTeX∫tan2u du=tanu−u+C\int \tan^2 u\,du = \tan u - u + C∫tan2udu=tanu−u+CVer detalhe →
FórmulaCopiar LaTeX∫cot2u du=−cotu−u+C\int \cot^2 u\,du = -\cot u - u + C∫cot2udu=−cotu−u+CVer detalhe →
A.2.2 Potências ímpares (n=3n = 3n=3)Copiar LaTeX∫sen3u du=−13(2+sen2u)cosu+C\int \operatorname{sen}^3 u\,du = -\frac{1}{3}\left(2 + \operatorname{sen}^2 u\right)\cos u + C∫sen3udu=−31(2+sen2u)cosu+CVer detalhe →
FórmulaCopiar LaTeX∫cos3u du=13(2+cos2u)sen u+C\int \cos^3 u\,du = \frac{1}{3}\left(2 + \cos^2 u\right)\operatorname{sen}\,u + C∫cos3udu=31(2+cos2u)senu+CVer detalhe →
FórmulaCopiar LaTeX∫tan3u du=12 tan2u+ln∣cosu∣+C\int \tan^3 u\,du = \frac{1}{2}\,\tan^2 u + \ln|\cos u| + C∫tan3udu=21tan2u+ln∣cosu∣+CVer detalhe →
FórmulaCopiar LaTeX∫cot3u du=−12 cot2u−ln∣sen u∣+C\int \cot^3 u\,du = -\frac{1}{2}\,\cot^2 u - \ln|\operatorname{sen}\,u| + C∫cot3udu=−21cot2u−ln∣senu∣+CVer detalhe →
FórmulaCopiar LaTeX∫sec3u du=12 secu tanu+12 ln∣secu+tanu∣+C\int \sec^3 u\,du = \frac{1}{2}\,\sec u\,\tan u + \frac{1}{2}\,\ln|\sec u + \tan u| + C∫sec3udu=21secutanu+21ln∣secu+tanu∣+CVer detalhe →
FórmulaCopiar LaTeX∫csc3u du=−12 cscu cotu+12 ln∣cscu−cotu∣+C\int \csc^3 u\,du = -\frac{1}{2}\,\csc u\,\cot u + \frac{1}{2}\,\ln|\csc u - \cot u| + C∫csc3udu=−21cscucotu+21ln∣cscu−cotu∣+CVer detalhe →
A.2.3 Fórmulas de reduçãoCopiar LaTeX∫sennu du=−1n senn−1u cosu+n−1n∫senn−2u du\int \operatorname{sen}^n u\,du = -\frac{1}{n}\,\operatorname{sen}^{n-1} u\,\cos u + \frac{n-1}{n}\int \operatorname{sen}^{n-2} u\,du∫sennudu=−n1senn−1ucosu+nn−1∫senn−2uduVer detalhe →
FórmulaCopiar LaTeX∫cosnu du=1n cosn−1u sen u+n−1n∫cosn−2u du\int \cos^n u\,du = \frac{1}{n}\,\cos^{n-1} u\,\operatorname{sen}\,u + \frac{n-1}{n}\int \cos^{n-2} u\,du∫cosnudu=n1cosn−1usenu+nn−1∫cosn−2uduVer detalhe →
FórmulaCopiar LaTeX∫tannu du=1n−1 tann−1u−∫tann−2u du\int \tan^n u\,du = \frac{1}{n-1}\,\tan^{n-1} u - \int \tan^{n-2} u\,du∫tannudu=n−11tann−1u−∫tann−2uduVer detalhe →