Trigonometric Integrals

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)

${\displaystyle \sin u}$ ${\displaystyle \cos u}$ ${\displaystyle \tan u}$

${\displaystyle \int \sin u\mathrm {d} u=-\cos u+C}$

${\displaystyle \int \cos u\mathrm {d} u=\sin u+C}$

${\displaystyle \int \tan u\mathrm {d} u=-\ln \left\vert \cos u\right\vert +C}$

${\displaystyle \int \sin ^{2}u\mathrm {d} u={\frac {1}{2}}(u-\sin u\cos u)+C}$

${\displaystyle \int \cos ^{2}u\mathrm {d} u={\frac {1}{2}}(u+\sin u\cos u)+C}$

${\displaystyle \int \tan ^{2}u\mathrm {d} u=-u+\tan u+C}$

${\displaystyle \int \sin ^{k}u\mathrm {d} u=-{\frac {\sin ^{k-1}u\cos u}{k}}+{\frac {k-1}{k}}\int \sin ^{k-2}u\mathrm {d} u}$

${\displaystyle \int \cos ^{k}u\mathrm {d} u={\frac {\cos ^{k-1}u\sin u}{k}}+{\frac {k-1}{k}}\int \cos ^{k-2}u\mathrm {d} u}$

${\displaystyle \int \tan ^{k}u\mathrm {d} u={\frac {\cos ^{k-1}u\sin u}{k}}+{\frac {k-1}{k}}\int \cos ^{k-2}u\mathrm {d} u}$

${\displaystyle \int u\sin u\mathrm {d} u=\sin u-u\cos u+C}$

${\displaystyle \int u\cos u\mathrm {d} u=\cos u+u\sin u+C}$

${\displaystyle \int u^{k}\sin u\mathrm {d} u=-u^{k}\cos u+k\int u^{k-1}\cos u\mathrm {d} u}$

${\displaystyle \int u^{k}\cos u\mathrm {d} u=u^{k}\sin u-k\int u^{k-1}\sin u\mathrm {d} u}$

${\displaystyle \int {\frac {1}{1\pm \sin u}}\mathrm {d} u=\tan u\pm \sec u+C}$

${\displaystyle \int {\frac {1}{1\pm \cos u}}\mathrm {d} u=-\cot u\pm \csc u+C}$

${\displaystyle \int {\frac {1}{1\pm \tan u}}\mathrm {d} u={\frac {1}{2}}(u\pm \ln \left\vert \cos u\pm \sin u\right\vert )+C}$

${\displaystyle \int {\frac {1}{\sin u\cos u}}\mathrm {d} u=\ln \left\vert \tan u\right\vert +C}$

${\displaystyle \cot u}$ ${\displaystyle \sec u}$ ${\displaystyle \csc u}$

${\displaystyle \int \cot u\mathrm {d} u=\ln \left\vert \sin u\right\vert +C}$

${\displaystyle \int \sec u\mathrm {d} u=\ln \left\vert \sec u+\tan u\right\vert +C}$

${\displaystyle \int \csc u\mathrm {d} u=\ln \left\vert \csc u-\cot u\right\vert +C}$

${\displaystyle \int \cot ^{2}u\mathrm {d} u=-u-\cot u+C}$

${\displaystyle \int \sec ^{2}u\mathrm {d} u=\tan u+C}$

${\displaystyle \int \csc ^{2}u\mathrm {d} u=-\cot u+C}$

${\displaystyle \int \cot ^{k}u\mathrm {d} u=-{\frac {\cot ^{k-1}u}{k-1}}\int \cot ^{k-2}u\mathrm {d} ,k\neq 1}$

${\displaystyle \int \sec ^{k}u\mathrm {d} u={\frac {\sec ^{k-2}u\tan u}{k-1}}+{\frac {k-2}{k-1}}\int \sec ^{k-2}u\mathrm {d} u,l\neq 1}$

${\displaystyle \int \csc ^{k}u\mathrm {d} u=-{\frac {\csc ^{k-2}u\tan u}{k-1}}+{\frac {k-2}{k-1}}\int \csc ^{k-2}u\mathrm {d} u,k/neq1}$

${\displaystyle \int {\frac {1}{1\pm \cot u}}\mathrm {d} u={\frac {1}{2}}(u\mp \ln \left\vert \sin u\pm \cos u\right\vert )+C}$

${\displaystyle \int {\frac {1}{1\pm \sec u}}\mathrm {d} u=u+\cot u\mp \csc u+C}$

${\displaystyle \int {\frac {1}{1\pm \csc }}\mathrm {d} u=u-\tan u\pm \sec u+C}$

Inverse Trig Functions

${\displaystyle \int \arcsin u\mathrm {d} u=u\arcsin u+{\sqrt {1-u^{2}}}+C}$

${\displaystyle \int \arccos u\mathrm {d} u=u\arccos u-{\sqrt {1-u^{2}}}+C}$

${\displaystyle \int \arctan u\mathrm {d} u=u\arctan u-\ln {\sqrt {1+u^{2}}}+C}$

${\displaystyle \int \operatorname {arccot} u\mathrm {d} u=u\operatorname {arccot} u+\ln {\sqrt {1+u^{2}}}+C}$

${\displaystyle \int \operatorname {arcsec} u\mathrm {d} u=u\operatorname {arcsec} u+\ln \left\vert u+{\sqrt {u^{2}-1}}\right\vert +C}$

${\displaystyle \int \operatorname {arccsc} u\mathrm {d} u=u\operatorname {arccsc} u+\ln \left\vert u+{\sqrt {u^{2}-1}}\right\vert +C}$

${\displaystyle e^{u}}$

${\displaystyle \int e^{u}\mathrm {d} u=e^{u}+C}$

${\displaystyle \int ue^{u}\mathrm {d} u=(u-1)e^{u}+C}$

${\displaystyle \int u^{k}e^{u}\mathrm {d} u=k\int u^{k-1}e^{u}\mathrm {d} u}$

${\displaystyle \int {\frac {1}{1+e^{u}}}\mathrm {d} u-u-\ln(1+e^{u})+C}$

${\displaystyle \int e^{au}\sin bu\mathrm {d} u={\frac {e^{au}}{a^{2}+b^{2}}}(a\sin bu-b\cos bu)+C}$

${\displaystyle \int e^{au}\cos bu\mathrm {d} u={\frac {e^{au}}{a^{2}+b^{2}}}(a\cos bu+b\sin bu)+C}$

${\displaystyle \ln u}$

${\displaystyle \int ]lnu\mathrm {d} u=u(-1+\ln u)+C}$

${\displaystyle \int u\ln u\mathrm {d} u={\frac {u^{2}}{4}}(-1+2\ln u)+C}$

${\displaystyle \int u^{k}\ln u\mathrm {d} u={\frac {e^{au}}{a^{2}+b^{2}}}(a\cos bu+b\sin bu)+C}$

${\displaystyle \int (\ln u)^{2}\mathrm {d} u=u[2-2\ln u+(\ln u)^{2}]+C}$

${\displaystyle \int (\ln u)^{k}\mathrm {d} u=u(\ln u)^{k}-k\int (\ln u)^{k-1}\mathrm {d} u}$

Hyperbolic Functions

${\displaystyle \int \cosh u\mathrm {d} u=\sinh u+C}$

${\displaystyle \int \sinh u\mathrm {d} u=\cosh u+C}$

${\displaystyle \int \operatorname {sech} ^{2}u\mathrm {d} u=\tanh u+C}$

${\displaystyle \int \operatorname {csch} ^{2}u\mathrm {d} u=-\coth u+C}$

${\displaystyle \int \operatorname {sech} u\tan u\mathrm {d} u=-\operatorname {sech} u+C}$

${\displaystyle \int \operatorname {csch} u\coth u\mathrm {d} u=-\operatorname {csch} u+C}$