High School Calculus/Trigonometric Integrals

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)

$\sin u$ $\cos u$ $\tan u$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\cot u$ $\sec u$ $\csc u$

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

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

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

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

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

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

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

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

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

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

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

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

Inverse Trig Functions

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

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

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

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

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

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

$e^{u}$

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

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

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

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

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

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

$\ln u$

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

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

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

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

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

Hyperbolic Functions

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

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

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

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

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

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