Home/Calculus II/Appendix A: extensive table of antiderivatives/A.2. Trigonometric integralsContentA.2. Trigonometric integralsA.2.1 Even powers (n=2n = 2n=2)Copy LaTeX∫sin2u du=12 u−14 sin2u+C\int \sin^2 u\,du = \frac{1}{2}\,u - \frac{1}{4}\,\sin 2u + C∫sin2udu=21u−41sin2u+CView detail →FormulaCopy LaTeX∫cos2u du=12 u+14 sin2u+C\int \cos^2 u\,du = \frac{1}{2}\,u + \frac{1}{4}\,\sin 2u + C∫cos2udu=21u+41sin2u+CView detail →FormulaCopy LaTeX∫tan2u du=tanu−u+C\int \tan^2 u\,du = \tan u - u + C∫tan2udu=tanu−u+CView detail →FormulaCopy LaTeX∫cot2u du=−cotu−u+C\int \cot^2 u\,du = -\cot u - u + C∫cot2udu=−cotu−u+CView detail →A.2.2 Odd powers (n=3n = 3n=3)Copy LaTeX∫sin3u du=−13(2+sin2u)cosu+C\int \sin^3 u\,du = -\frac{1}{3}\left(2 + \sin^2 u\right)\cos u + C∫sin3udu=−31(2+sin2u)cosu+CView detail →FormulaCopy LaTeX∫cos3u du=13(2+cos2u)sin u+C\int \cos^3 u\,du = \frac{1}{3}\left(2 + \cos^2 u\right)\sin\,u + C∫cos3udu=31(2+cos2u)sinu+CView detail →FormulaCopy 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∣+CView detail →FormulaCopy LaTeX∫cot3u du=−12 cot2u−ln∣sin u∣+C\int \cot^3 u\,du = -\frac{1}{2}\,\cot^2 u - \ln|\sin\,u| + C∫cot3udu=−21cot2u−ln∣sinu∣+CView detail →FormulaCopy 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∣+CView detail →FormulaCopy 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∣+CView detail →A.2.3 Reduction formulasCopy LaTeX∫sinnu du=−1n sinn−1u cosu+n−1n∫sinn−2u du\int \sin^n u\,du = -\frac{1}{n}\,\sin^{n-1} u\,\cos u + \frac{n-1}{n}\int \sin^{n-2} u\,du∫sinnudu=−n1sinn−1ucosu+nn−1∫sinn−2uduView detail →FormulaCopy LaTeX∫cosnu du=1n cosn−1u sin u+n−1n∫cosn−2u du\int \cos^n u\,du = \frac{1}{n}\,\cos^{n-1} u\,\sin\,u + \frac{n-1}{n}\int \cos^{n-2} u\,du∫cosnudu=n1cosn−1usinu+nn−1∫cosn−2uduView detail →FormulaCopy 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−2uduView detail →
A.2.1 Even powers (n=2n = 2n=2)Copy LaTeX∫sin2u du=12 u−14 sin2u+C\int \sin^2 u\,du = \frac{1}{2}\,u - \frac{1}{4}\,\sin 2u + C∫sin2udu=21u−41sin2u+CView detail →
FormulaCopy LaTeX∫cos2u du=12 u+14 sin2u+C\int \cos^2 u\,du = \frac{1}{2}\,u + \frac{1}{4}\,\sin 2u + C∫cos2udu=21u+41sin2u+CView detail →
FormulaCopy LaTeX∫tan2u du=tanu−u+C\int \tan^2 u\,du = \tan u - u + C∫tan2udu=tanu−u+CView detail →
FormulaCopy LaTeX∫cot2u du=−cotu−u+C\int \cot^2 u\,du = -\cot u - u + C∫cot2udu=−cotu−u+CView detail →
A.2.2 Odd powers (n=3n = 3n=3)Copy LaTeX∫sin3u du=−13(2+sin2u)cosu+C\int \sin^3 u\,du = -\frac{1}{3}\left(2 + \sin^2 u\right)\cos u + C∫sin3udu=−31(2+sin2u)cosu+CView detail →
FormulaCopy LaTeX∫cos3u du=13(2+cos2u)sin u+C\int \cos^3 u\,du = \frac{1}{3}\left(2 + \cos^2 u\right)\sin\,u + C∫cos3udu=31(2+cos2u)sinu+CView detail →
FormulaCopy 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∣+CView detail →
FormulaCopy LaTeX∫cot3u du=−12 cot2u−ln∣sin u∣+C\int \cot^3 u\,du = -\frac{1}{2}\,\cot^2 u - \ln|\sin\,u| + C∫cot3udu=−21cot2u−ln∣sinu∣+CView detail →
FormulaCopy 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∣+CView detail →
FormulaCopy 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∣+CView detail →
A.2.3 Reduction formulasCopy LaTeX∫sinnu du=−1n sinn−1u cosu+n−1n∫sinn−2u du\int \sin^n u\,du = -\frac{1}{n}\,\sin^{n-1} u\,\cos u + \frac{n-1}{n}\int \sin^{n-2} u\,du∫sinnudu=−n1sinn−1ucosu+nn−1∫sinn−2uduView detail →
FormulaCopy LaTeX∫cosnu du=1n cosn−1u sin u+n−1n∫cosn−2u du\int \cos^n u\,du = \frac{1}{n}\,\cos^{n-1} u\,\sin\,u + \frac{n-1}{n}\int \cos^{n-2} u\,du∫cosnudu=n1cosn−1usinu+nn−1∫cosn−2uduView detail →
FormulaCopy 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−2uduView detail →