Anexo:Integrales de funciones trigonométricas

La siguiente es una lista de integrales de funciones trigonométricas y su correspondiente simplificación. La letra c representa una constante numérica.

Integrales que contienen solamente sen

$\int \mathrm{sen}cxdx=-\frac{1}{c}\cos cx$

$\int {\mathrm{sen}}^{n}cxdx=-\frac{{\mathrm{sen}}^{n-1}cx\cos cx}{nc}+\frac{n-1}{n}\int {\mathrm{sen}}^{n-2}cxdx\text{(para }n>0\text{)}$

$\int x\mathrm{sen}cxdx=\frac{\mathrm{sen}cx}{c^2}-\frac{x\cos cx}{c}$

$\int x^n\mathrm{sen}cxdx=-\frac{x^n}{c}\cos cx+\frac{n}{c}\int x^{n-1}\cos cxdx\text{(para }n>0\text{)}$

$\int \frac{\mathrm{sen}cx}{x}dx=\sum _{i=0}^{\mathrm{\infty }}(-1{)}^i\frac{(cx{)}^{2i+1}}{(2i+1)\cdot (2i+1)!}$

$\int \frac{\mathrm{sen}cx}{x^n}dx=-\frac{\mathrm{sen}cx}{(n-1)x^{n-1}}+\frac{c}{n-1}\int \frac{\cos cx}{x^{n-1}}dx$

$\int \frac{dx}{\mathrm{sen}cx}=\frac{1}{c}\ln |\tan \frac{cx}{2}|$

$\int \frac{dx}{{\mathrm{sen}}^{n}cx}=\frac{\cos cx}{c(1-n){\mathrm{sen}}^{n-1}cx}+\frac{n-2}{n-1}\int \frac{dx}{{\mathrm{sen}}^{n-2}cx}\text{(para }n>1\text{)}$

$\int \frac{dx}{1\pm \mathrm{sen}cx}=\frac{1}{c}\tan (\frac{cx}{2}\mp \frac{\pi }{4})$

$\int \frac{xdx}{1-\mathrm{sen}cx}=\frac{x}{c}\cot (\frac{\pi }{4}-\frac{cx}{2})+\frac{2}{c^2}\ln |\mathrm{sen}(\frac{\pi }{4}-\frac{cx}{2})|$

$\int \frac{\mathrm{sen}cxdx}{1\pm \mathrm{sen}cx}=\pm x+\frac{1}{c}\tan (\frac{\pi }{4}\mp \frac{cx}{2})$

$\int \mathrm{sen}{c}_{1}x\mathrm{sen}{c}_{2}xdx=\frac{\mathrm{sen}(c_1-c_2)x}{2(c_1-c_2)}-\frac{\mathrm{sen}(c_1+c_2)x}{2(c_1+c_2)}\text{(para }|c_1|\ne |c_2|\text{)}$

$\int sen\mathrm{(}{ax}^2\mathrm{+}bx\mathrm{+}c\mathrm{)}dx\mathrm{=}\{\begin{array}{cc}& \sqrt{a}\sqrt{\frac{\mathit{\pi }}{2}}\cos \left(\frac{b^2\mathrm{-}4ac}{4a}\right)S\left(\frac{2ax\mathrm{+}b}{\sqrt{2a\mathit{\pi }}}\right)\mathrm{+}\sqrt{a}\sqrt{\frac{\mathit{\pi }}{2}}\mathrm{sen}\left(\frac{b^2\mathrm{-}4ac}{4a}\right)C\left(\frac{2ax\mathrm{+}b}{\sqrt{2a\mathit{\pi }}}\right)para{b}^2\mathrm{-}4ac>0\\ & \sqrt{a}\sqrt{\frac{\mathit{\pi }}{2}}\cos \left(\frac{b^2\mathrm{-}4ac}{4a}\right)S\left(\frac{2ax\mathrm{+}b}{\sqrt{2a\mathit{\pi }}}\right)\mathrm{-}\sqrt{a}\sqrt{\frac{\mathit{\pi }}{2}}\mathrm{sen}\left(\frac{b^2\mathrm{-}4ac}{4a}\right)C\left(\frac{2ax\mathrm{+}b}{\sqrt{2a\mathit{\pi }}}\right)para{b}^2\mathrm{-}4ac<0\end{array}ya╱\mathrm{=}0$

Sean S y C integrales de Fresnel

Integrales que contienen solamente cos

$\int \cos cxdx=\frac{1}{c}\mathrm{sen}cx$

$\int {\cos }^{n}cxdx=\frac{{\cos }^{n-1}cx\mathrm{sen}cx}{nc}+\frac{n-1}{n}\int {\cos }^{n-2}cxdx\text{(para }n>0\text{)}$

$\int x\cos cxdx=\frac{\cos cx}{c^2}+\frac{x\mathrm{sen}cx}{c}$

$\int x^n\cos cxdx=\frac{x^n\mathrm{sen}cx}{c}-\frac{n}{c}\int x^{n-1}\mathrm{sen}cxdx$

$\int \frac{\cos cx}{x}dx=\ln |cx|+\sum _{i=1}^{\mathrm{\infty }}(-1{)}^i\frac{(cx{)}^{2i}}{2i\cdot (2i)!}$

$\int \frac{\cos cx}{x^n}dx=-\frac{\cos cx}{(n-1)x^{n-1}}-\frac{c}{n-1}\int \frac{\mathrm{sen}cx}{x^{n-1}}dx\text{(para }n\ne 1\text{)}$

$\int \frac{dx}{\cos cx}=\frac{1}{c}\ln |\tan (\frac{cx}{2}+\frac{\pi }{4})|$

$\int \frac{dx}{{\cos }^{n}cx}=\frac{\mathrm{sen}cx}{c(n-1)co{s}^{n-1}cx}+\frac{n-2}{n-1}\int \frac{dx}{{\cos }^{n-2}cx}\text{(para }n>1\text{)}$

$\int \frac{dx}{1+\cos cx}=\frac{1}{c}\tan \frac{cx}{2}$

$\int \frac{dx}{1-\cos cx}=-\frac{1}{c}\cot \frac{cx}{2}$

$\int \frac{xdx}{1+\cos cx}=\frac{x}{c}\tan cx2+\frac{2}{c^2}\ln |\cos \frac{cx}{2}|$

$\int \frac{xdx}{1-\cos cx}=-\frac{x}{x}\cot cx2+\frac{2}{c^2}\ln |\mathrm{sen}\frac{cx}{2}|$

$\int \frac{\cos cxdx}{1+\cos cx}=x-\frac{1}{c}\tan \frac{cx}{2}$

$\int \frac{\cos cxdx}{1-\cos cx}=-x-\frac{1}{c}\cot \frac{cx}{2}$

$\int \cos c_{1}x\cos c_{2}xdx=\frac{\mathrm{sen}(c_1-c_2)x}{2(c_1-c_2)}+\frac{\mathrm{sen}(c_1+c_2)x}{2(c_1+c_2)}\text{(para }|c_1|\ne |c_2|\text{)}$

Integrales que contienen solamente tan

$\int \tan cxdx=-\frac{1}{c}\ln |\cos cx|$

$\int {\tan }^{n}cxdx=\frac{1}{c(n-1)}{\tan }^{n-1}cx-\int {\tan }^{n-2}cxdx\text{(para }n\ne 1\text{)}$

$\int \frac{dx}{\tan cx+1}=\frac{x}{2}+\frac{1}{2c}\ln |\mathrm{sen}cx+\cos cx|$

$\int \frac{dx}{\tan cx-1}=-\frac{x}{2}+\frac{1}{2c}\ln |\mathrm{sen}cx-\cos cx|$

$\int \frac{\tan cxdx}{\tan cx+1}=\frac{x}{2}-\frac{1}{2c}\ln |\mathrm{sen}cx+\cos cx|$

$\int \frac{\tan cxdx}{\tan cx-1}=\frac{x}{2}+\frac{1}{2c}\ln |\mathrm{sen}cx-\cos cx|$

Integrales que contienen solamente cot

$\int \cot cxdx=\frac{1}{c}\ln |\mathrm{sen}cx|$

$\int {\cot }^{n}cxdx=-\frac{1}{c(n-1)}{\cot }^{n-1}cx-\int {\cot }^{n-2}cxdx\text{(para )}n\ne 1\text{)}$

$\int \frac{dx}{1+\cot cx}=\int \frac{\tan cxdx}{\tan cx+1}$

$\int \frac{dx}{1-\cot cx}=\int \frac{\tan cxdx}{\tan cx-1}$

Integrales que contienen sen y cos

$\int \frac{dx}{\cos cx\pm \mathrm{sen}cx}=\frac{1}{c\sqrt{2}}\ln |\tan (\frac{cx}{2}\pm \frac{\pi }{8})|$

$\int \frac{dx}{(\cos cx\pm \mathrm{sen}cx{)}^2}=\frac{1}{2c}\tan (cx\mp \frac{\pi }{4})$

$\int \frac{\cos cxdx}{\cos cx+\mathrm{sen}cx}=\frac{x}{2}+\frac{1}{2c}\ln |\mathrm{sen}cx+\cos cx|$

$\int \frac{\cos cxdx}{\cos cx-\mathrm{sen}cx}=\frac{x}{2}-\frac{1}{2c}\ln |\mathrm{sen}cx-\cos cx|$

$\int \frac{\mathrm{sen}cxdx}{\cos cx+\mathrm{sen}cx}=\frac{x}{2}-\frac{1}{2c}\ln |\mathrm{sen}cx+\cos cx|$

$\int \frac{\mathrm{sen}cxdx}{\cos cx-\mathrm{sen}cx}=-\frac{x}{2}-\frac{1}{2c}\ln |\mathrm{sen}cx-\cos cx|$

$\int \frac{\cos cxdx}{\mathrm{sen}cx(1+\cos cx)}=-\frac{1}{4c}{\tan }^2\frac{cx}{2}+\frac{1}{2c}\ln |\tan \frac{cx}{2}|$

$\int \frac{\cos cxdx}{\mathrm{sen}cx(1+-\cos cx)}=-\frac{1}{4c}{\cot }^2\frac{cx}{2}-\frac{1}{2c}\ln |\tan \frac{cx}{2}|$

$\int \frac{\mathrm{sen}cxdx}{\cos cx(1+\mathrm{sen}cx)}=\frac{1}{4c}{\cot }^2(\frac{cx}{2}+\frac{\pi }{4})+\frac{1}{2c}\ln |\tan (\frac{cx}{2}+\frac{\pi }{4})|$

$\int \frac{\mathrm{sen}cxdx}{\cos cx(1-\mathrm{sen}cx)}=\frac{1}{4c}{\tan }^2(\frac{cx}{2}+\frac{\pi }{4})-\frac{1}{2c}\ln |\tan (\frac{cx}{2}+\frac{\pi }{4})|$

$\int \mathrm{sen}cx\cos cxdx=\frac{1}{2c}{\mathrm{sen}}^{2}cx$

$\int \mathrm{sen}{c}_{1}x\cos c_{2}xdx=-\frac{\cos (c_1+c_2)x}{2(c_1+c_2)}-\frac{\cos (c_1-c_2)x}{2(c_1-c_2)}\text{(para }|c_1|\ne |c_2|\text{)}$

$\int {\mathrm{sen}}^{n}cx\cos cxdx=\frac{1}{c(n+1)}{\mathrm{sen}}^{n+1}cx\text{(para }n\ne 1\text{)}$

$\int \mathrm{sen}cx{\cos }^{n}cxdx=-\frac{1}{c(n+1)}{\cos }^{n+1}cx\text{(para }n\ne 1\text{)}$

$\int {\mathrm{sen}}^{n}cx{\cos }^{m}cxdx=-\frac{{\mathrm{sen}}^{n-1}cx{\cos }^{m+1}cx}{c(n+m)}+\frac{n-1}{n+m}\int {\mathrm{sen}}^{n-2}cx{\cos }^{m}cxdx\text{(para }m,n>0\text{)}$

también: $\int {\mathrm{sen}}^{n}cx{\cos }^{m}cxdx=\frac{{\mathrm{sen}}^{n+1}cx{\cos }^{m-1}cx}{c(n+m)}+\frac{m-1}{n+m}\int {\mathrm{sen}}^{n}cx{\cos }^{m-2}cxdx\text{(para }m,n>0\text{)}$

$\int \frac{dx}{\mathrm{sen}cx\cos cx}=\frac{1}{c}\ln |\tan cx|$

$\int \frac{dx}{\mathrm{sen}cx{\cos }^{n}cx}=\frac{1}{c(n-1){\cos }^{n-1}cx}+\int \frac{dx}{\mathrm{sen}cx{\cos }^{n-2}cx}\text{(para }n\ne 1\text{)}$

$\int \frac{dx}{{\mathrm{sen}}^{n}cx\cos cx}=-\frac{1}{c(n-1){\mathrm{sen}}^{n-1}cx}+\int \frac{dx}{{\mathrm{sen}}^{n-2}cx\cos cx}\text{(para }n\ne 1\text{)}$

$\int \frac{\mathrm{sen}cxdx}{{\cos }^{n}cx}=\frac{1}{c(n-1){\cos }^{n-1}cx}\text{(para }n\ne 1\text{)}$

$\int \frac{{\mathrm{sen}}^{2}cxdx}{\cos cx}=-\frac{1}{c}\mathrm{sen}cx+\frac{1}{c}\ln |\tan (\frac{\pi }{4}+\frac{cx}{2})|$

$\int \frac{{\mathrm{sen}}^{2}cxdx}{{\cos }^{n}cx}=\frac{\mathrm{sen}cx}{c(n-1){\cos }^{n-1}cx}-\frac{1}{n-1}\int \frac{dx}{{\cos }^{n-2}cx}\text{(para }n\ne 1\text{)}$

$\int \frac{{\mathrm{sen}}^{n}cxdx}{\cos cx}=-\frac{{\mathrm{sen}}^{n-1}cx}{c(n-1)}+\int \frac{{\mathrm{sen}}^{n-2}cxdx}{\cos cx}\text{(for }n\ne 1\text{)}$

$\int \frac{se{n}^{n}cxdx}{{\cos }^{m}cx}=\frac{{\mathrm{sen}}^{n+1}cx}{c(m-1){\cos }^{m-1}cx}-\frac{n-m+2}{m-1}\int \frac{{\mathrm{sen}}^{n}cxdx}{{\cos }^{m-2}cx}\text{(para }m\ne 1\text{)}$

también: $\int \frac{se{n}^{n}cxdx}{{\cos }^{m}cx}=-\frac{{\mathrm{sen}}^{n-1}cx}{c(n-m){\cos }^{m-1}cx}+\frac{n-1}{n-m}\int \frac{{\mathrm{sen}}^{n-2}cxdx}{{\cos }^{m}cx}\text{(para }m\ne n\text{)}$

también: $\int \frac{se{n}^{n}cxdx}{{\cos }^{m}cx}=\frac{{\mathrm{sen}}^{n-1}cx}{c(m-1){\cos }^{m-1}cx}-\frac{n-1}{n-1}\int \frac{{\mathrm{sen}}^{n-1}cxdx}{{\cos }^{m-2}cx}\text{(para }m\ne 1\text{)}$

$\int \frac{\cos cxdx}{{\mathrm{sen}}^{n}cx}=-\frac{1}{c(n-1){\mathrm{sen}}^{n-1}cx}\text{(para }n\ne 1\text{)}$

$\int \frac{{\cos }^{2}cxdx}{\mathrm{sen}cx}=\frac{1}{c}(\cos cx+\ln |\tan \frac{cx}{2}|)$

$\int \frac{{\cos }^{2}cxdx}{{\mathrm{sen}}^{n}cx}=-\frac{1}{n-1}(\frac{\cos cx}{c{\mathrm{sen}}^{n-1}cx)}+\int \frac{dx}{{\mathrm{sen}}^{n-2}cx})\text{(para }n\ne 1\text{)}$

$\int \frac{{\cos }^{n}cxdx}{{\mathrm{sen}}^{m}cx}=-\frac{{\cos }^{n+1}cx}{c(m-1){\mathrm{sen}}^{m-1}cx}-\frac{n-m-2}{m-1}\int \frac{co{s}^{n}cxdx}{{\mathrm{sen}}^{m-2}cx}\text{(para }m\ne 1\text{)}$

también: $\int \frac{{\cos }^{n}cxdx}{{\mathrm{sen}}^{m}cx}=\frac{{\cos }^{n-1}cx}{c(n-m){\mathrm{sen}}^{m-1}cx}+\frac{n-1}{n-m}\int \frac{co{s}^{n-2}cxdx}{{\mathrm{sen}}^{m}cx}\text{(para }m\ne n\text{)}$

también: $\int \frac{{\cos }^{n}cxdx}{{\mathrm{sen}}^{m}cx}=-\frac{{\cos }^{n-1}cx}{c(m-1){\mathrm{sen}}^{m-1}cx}-\frac{n-1}{m-1}\int \frac{co{s}^{n-2}cxdx}{{\mathrm{sen}}^{m-2}cx}\text{(para }m\ne 1\text{)}$

Integrales que contienen sen y tan

$\int \mathrm{sen}cx\tan cxdx=\frac{1}{c}(\ln |\sec cx+\tan cx|-\mathrm{sen}cx)$

$\int \frac{{\tan }^{n}cxdx}{{\mathrm{sen}}^{2}cx}=\frac{1}{f}ffc(n-1){\tan }^{n-1}(cx)\text{(para }n\ne 1\text{)}$

Integrales que contienen cos y tan

$\int \frac{{\tan }^{n}cxdx}{{\cos }^{2}cx}=\frac{1}{c(n+1)}{\tan }^{n+1}cx\text{(para }n\ne 1\text{)}$

Integrales que contienen sen y cot

$\int \frac{{\cot }^{n}cxdx}{se{n}^{2}cx}=\frac{1}{c(n+1)}{\cot }^{n+1}cx\text{(para }n\ne 1\text{)}$

Integrales que contienen cos y cot

$\int \frac{{\cot }^{n}cxdx}{{\cos }^{2}cx}=\frac{1}{c(1-n)}{\tan }^{1-n}cx\text{(para }n\ne 1\text{)}$

Integrales que contienen tan y cot

$\int \cot cx\tan cxdx=x$

Integrales que contienen sec

$\int {\sec }^{2}xdx=\tan x+c$

$\int \sec xdx=\ln |\sec x+\tan x|+c$