There are a few different types of integrals using trigonometric functions. I will split it into a few different sections. Those involving sine, cosine, and tangent. From there I will cover cotangent, secant, and cosecant. Then I will cover the inverse functions, functions involving e, ln, and finally hyperbolic functions.
Something to keep in mind is that the variable used in these functions are denoted by u. (see substitution section)
![{\displaystyle \tan u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6d99e114ec0fc2a771ec70ceeed54a70a62b404)
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![{\displaystyle \int \cos u\mathrm {d} u=\sin u+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53f6f7881edc8d334b52d331f73a6ddf4c48117b)
![{\displaystyle \int \tan u\mathrm {d} u=-\ln \left\vert \cos u\right\vert +C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82154efbb2f2f17a3a2662f91be33133a6fc0fa9)
![{\displaystyle \int \sin ^{2}u\mathrm {d} u={\frac {1}{2}}(u-\sin u\cos u)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/286196839c915b00fc6e23bfb7cb428e3ce40f89)
![{\displaystyle \int \cos ^{2}u\mathrm {d} u={\frac {1}{2}}(u+\sin u\cos u)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c92d5d82dfb72d1ea43891f8377e846fcb7a6e6)
![{\displaystyle \int \tan ^{2}u\mathrm {d} u=-u+\tan u+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c478daaaa9bcc571258e3ce84a3720114dfd9304)
![{\displaystyle \int \sin ^{k}u\mathrm {d} u=-{\frac {\sin ^{k-1}u\cos u}{k}}+{\frac {k-1}{k}}\int \sin ^{k-2}u\mathrm {d} u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea60dc9e3ea05598960001b0d3632276fa250566)
![{\displaystyle \int \cos ^{k}u\mathrm {d} u={\frac {\cos ^{k-1}u\sin u}{k}}+{\frac {k-1}{k}}\int \cos ^{k-2}u\mathrm {d} u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b302e918eae5a809cfb5e8f7ec4cbac0588fae8)
![{\displaystyle \int \tan ^{k}u\mathrm {d} u={\frac {\cos ^{k-1}u\sin u}{k}}+{\frac {k-1}{k}}\int \cos ^{k-2}u\mathrm {d} u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e209a42d95e0f7a389c7e7385593e8d90cc18778)
![{\displaystyle \int u\sin u\mathrm {d} u=\sin u-u\cos u+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2d075595c92f284622156970a3013fd9831bc7f)
![{\displaystyle \int u\cos u\mathrm {d} u=\cos u+u\sin u+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60f5f30ca842cc4631a42eecf35f10d6dec3d6b3)
![{\displaystyle \int u^{k}\sin u\mathrm {d} u=-u^{k}\cos u+k\int u^{k-1}\cos u\mathrm {d} u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13a8a853a7e5ce89a5334fac5e330d7f6a6460a8)
![{\displaystyle \int u^{k}\cos u\mathrm {d} u=u^{k}\sin u-k\int u^{k-1}\sin u\mathrm {d} u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6410c71f7261c5ed71ce60e1166285a575f694d2)
![{\displaystyle \int {\frac {1}{1\pm \sin u}}\mathrm {d} u=\tan u\pm \sec u+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d27560c578d1755493db6e49bd5ff061969a0c)
![{\displaystyle \int {\frac {1}{1\pm \cos u}}\mathrm {d} u=-\cot u\pm \csc u+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc19801ec448545b98b80404f416b6167927063a)
![{\displaystyle \int {\frac {1}{1\pm \tan u}}\mathrm {d} u={\frac {1}{2}}(u\pm \ln \left\vert \cos u\pm \sin u\right\vert )+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9f3e2d971dba747ecc4dbfaccd7860d069b69a9)
![{\displaystyle \int {\frac {1}{\sin u\cos u}}\mathrm {d} u=\ln \left\vert \tan u\right\vert +C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a8218a9d72d91004d8aa2ba23a403a577d99ac0)
![{\displaystyle \csc u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d12c5f4c2bfcbb7493995d6a556a73eeafd38d06)
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![{\displaystyle \int \sec u\mathrm {d} u=\ln \left\vert \sec u+\tan u\right\vert +C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a997fa7c4123eef8c09144665c611aeef6ac3280)
![{\displaystyle \int \csc u\mathrm {d} u=\ln \left\vert \csc u-\cot u\right\vert +C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed51cd437e51ed5df9e3a7f8253b2b5bfd96af21)
![{\displaystyle \int \cot ^{2}u\mathrm {d} u=-u-\cot u+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b6de8ad9f5dcae015994f02e8bb8e9a4f52ee712)
![{\displaystyle \int \sec ^{2}u\mathrm {d} u=\tan u+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7eb96077b414ef969dc5b076f7e8c17b96ef6b27)
![{\displaystyle \int \csc ^{2}u\mathrm {d} u=-\cot u+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5c24ad0a49e46e50a0b236f765f198b25abf093)
![{\displaystyle \int \cot ^{k}u\mathrm {d} u=-{\frac {\cot ^{k-1}u}{k-1}}\int \cot ^{k-2}u\mathrm {d} ,k\neq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18b2e0f791b2a26b4d05e4918eb40b5666fdaf3d)
![{\displaystyle \int \sec ^{k}u\mathrm {d} u={\frac {\sec ^{k-2}u\tan u}{k-1}}+{\frac {k-2}{k-1}}\int \sec ^{k-2}u\mathrm {d} u,l\neq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27b1444d3e55b41f0adfce266b3dffffba00c3ed)
![{\displaystyle \int \csc ^{k}u\mathrm {d} u=-{\frac {\csc ^{k-2}u\tan u}{k-1}}+{\frac {k-2}{k-1}}\int \csc ^{k-2}u\mathrm {d} u,k/neq1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de8c786cf7d96102e49be75fa5548a33c547b909)
![{\displaystyle \int {\frac {1}{1\pm \cot u}}\mathrm {d} u={\frac {1}{2}}(u\mp \ln \left\vert \sin u\pm \cos u\right\vert )+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92188feff437759e88a0fa1d81e2f34659207863)
![{\displaystyle \int {\frac {1}{1\pm \sec u}}\mathrm {d} u=u+\cot u\mp \csc u+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62548fd8cf72a24447f0cafcf142649dda111a63)
![{\displaystyle \int {\frac {1}{1\pm \csc }}\mathrm {d} u=u-\tan u\pm \sec u+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7082d5513a5d843b105ece5c126a7e9f1db86f07)
![{\displaystyle \int \arccos u\mathrm {d} u=u\arccos u-{\sqrt {1-u^{2}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0afb5eedda89f5e44e04e43076fc8d9f4914ce62)
![{\displaystyle \int \arctan u\mathrm {d} u=u\arctan u-\ln {\sqrt {1+u^{2}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9bb94bea296daa0dbaecf09f89dca20c0de1a29)
![{\displaystyle \int \operatorname {arccot} u\mathrm {d} u=u\operatorname {arccot} u+\ln {\sqrt {1+u^{2}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa465967e2fe2be9af09740ca70406c135cb16ad)
![{\displaystyle \int \operatorname {arcsec} u\mathrm {d} u=u\operatorname {arcsec} u+\ln \left\vert u+{\sqrt {u^{2}-1}}\right\vert +C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/342ed31c18778f68f117bfe8562ecd9ac133f69b)
![{\displaystyle \int \operatorname {arccsc} u\mathrm {d} u=u\operatorname {arccsc} u+\ln \left\vert u+{\sqrt {u^{2}-1}}\right\vert +C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2027dc1083283e97be811c2da63dc936700c757)
![{\displaystyle \int ue^{u}\mathrm {d} u=(u-1)e^{u}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/501a095c013879dcb1c96fe6687bbd5ea6a5b773)
![{\displaystyle \int u^{k}e^{u}\mathrm {d} u=k\int u^{k-1}e^{u}\mathrm {d} u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35332a6e4256528fad20da63a9e6715e9aab9bab)
![{\displaystyle \int {\frac {1}{1+e^{u}}}\mathrm {d} u-u-\ln(1+e^{u})+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef6ad620c066333db70f947b67ebd9096e613b9a)
![{\displaystyle \int e^{au}\sin bu\mathrm {d} u={\frac {e^{au}}{a^{2}+b^{2}}}(a\sin bu-b\cos bu)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd7e489b99cc4c15c89f3aac31df2b91039fad07)
![{\displaystyle \int e^{au}\cos bu\mathrm {d} u={\frac {e^{au}}{a^{2}+b^{2}}}(a\cos bu+b\sin bu)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a83fa4d90c8c43afcfef8ff821ec47c2b20bc85)
![{\displaystyle \int u\ln u\mathrm {d} u={\frac {u^{2}}{4}}(-1+2\ln u)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11e6c45a879505359ab7445e297231b8aab6292f)
![{\displaystyle \int u^{k}\ln u\mathrm {d} u={\frac {e^{au}}{a^{2}+b^{2}}}(a\cos bu+b\sin bu)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/413fb7f654d02d7bdeab79ee9b7ed18689ab1522)
![{\displaystyle \int (\ln u)^{2}\mathrm {d} u=u[2-2\ln u+(\ln u)^{2}]+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a06ecae7479c489698af96690a749d9c6c5dcb03)
![{\displaystyle \int (\ln u)^{k}\mathrm {d} u=u(\ln u)^{k}-k\int (\ln u)^{k-1}\mathrm {d} u}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8ca1ab59e9f7f63720a1b90a7a5dd6ee534af27)
![{\displaystyle \int \sinh u\mathrm {d} u=\cosh u+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d54cec3c7eac685995adaf126bcaf0723d295ec4)
![{\displaystyle \int \operatorname {sech} ^{2}u\mathrm {d} u=\tanh u+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18bba69745874311d2c002351bb074c03f9f0063)
![{\displaystyle \int \operatorname {csch} ^{2}u\mathrm {d} u=-\coth u+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22ee7604b5533f1fed6a1065116aee7dbf693d4e)
![{\displaystyle \int \operatorname {sech} u\tan u\mathrm {d} u=-\operatorname {sech} u+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae616b270bb10b6c3cccce39185899b6686bd755)
![{\displaystyle \int \operatorname {csch} u\coth u\mathrm {d} u=-\operatorname {csch} u+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41ddbfdb7d421920bb1ab605216c5d764e059e15)