Popis integrala trigonometrijskih funkcija

Slijedi popis integrala (antiderivacija funkcija) trigonometrijskih funkcija. Za potpun popis integrala funkcija, pogledati tablica integrala i popis integrala. Vidjeti također: trigonometrijski integral.

Za konstantu c se pretpostavlja da je različita od nule.

Integrali trigonometrijskih funkcija koje sadrže samo sin

Pri čemu je c konstanta:

$\int \sin cxdx=-\frac{1}{c}\cos cx+C$

$\int |sinx|dx=-\cos x+C$

$\int {\sin }^{n}cxdx=-\frac{{\sin }^{n-1}cx\cos cx}{nc}+\frac{n-1}{n}\int {\sin }^{n-2}cxdx+C\text{(za }n>0\text{)}$

$\int {\sin }^{2}cxdx=\frac{x}{2}-\frac{1}{4c}\sin 2cx+C$

$\int \sqrt{1-\sin x}dx=\int \sqrt{\mathrm{cvs}x}dx=2\frac{\cos \frac{x}{2}+\sin \frac{x}{2}}{\cos \frac{x}{2}-\sin \frac{x}{2}}\sqrt{\mathrm{cvs}x}=2\sqrt{1+\sin x}+C$

gdje je cvs{x} funkcija koversinus.

$\int x\sin cxdx=\frac{\sin cx}{c^2}-\frac{x\cos cx}{c}+C$

$\int x^n\sin cxdx=-\frac{x^n}{c}\cos cx+\frac{n}{c}\int x^{n-1}\cos cxdx+C\text{(za }n>0\text{)}$

$\underset{\frac{-a}{2}}{\overset{\frac{a}{2}}{\int }}{x}^2{\sin }^2\frac{n\pi x}{a}dx=\frac{a^3(n^2{\pi }^2-6)}{24n^2{\pi }^2}\text{(za }n=2,4,6...\text{)}$

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

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

$\int \frac{dx}{\sin cx}=\frac{1}{c}\ln |\mathrm{tg}\frac{cx}{2}|+C$

$\int \frac{dx}{{\sin }^{n}cx}=\frac{\cos cx}{c(1-n){\sin }^{n-1}cx}+\frac{n-2}{n-1}\int \frac{dx}{{\sin }^{n-2}cx}+C\text{(za }n>1\text{)}$

$\int \frac{dx}{1\pm \sin cx}=\frac{1}{c}\mathrm{tg}(\frac{cx}{2}\mp \frac{\pi }{4})+C$

$\int \frac{xdx}{1+\sin cx}=\frac{x}{c}\mathrm{tg}(\frac{cx}{2}-\frac{\pi }{4})+\frac{2}{c^2}\ln |\cos (\frac{cx}{2}-\frac{\pi }{4})|+C$

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

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

$\int \sin c_{1}x\sin c_{2}xdx=\frac{\sin (c_1-c_2)x}{2(c_1-c_2)}-\frac{\sin (c_1+c_2)x}{2(c_1+c_2)}+C\text{(za }|c_1|\ne |c_2|\text{)}$

Integrali trigonometrijskih funkcija koje sadrže samo cos

$\int \cos cxdx=\frac{1}{c}\sin cx+C$

$\int {\cos }^{n}cxdx=\frac{{\cos }^{n-1}cx\sin cx}{nc}+\frac{n-1}{n}\int {\cos }^{n-2}cxdx+C\text{(za }n>0\text{)}$

$\int x\cos cxdx=\frac{\cos cx}{c^2}+\frac{x\sin cx}{c}+C$

$\int x^n\cos cxdx=\frac{x^n\sin cx}{c}-\frac{n}{c}\int x^{n-1}\sin cxdx+C$

$\underset{\frac{-a}{2}}{\overset{\frac{a}{2}}{\int }}{x}^2{\cos }^2\frac{n\pi x}{a}dx=\frac{a^3(n^2{\pi }^2-6)}{24n^2{\pi }^2}\text{(za }n=1,3,5...\text{)}$

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

$\int \frac{\cos cx}{x^n}dx=-\frac{\cos cx}{(n-1)x^{n-1}}-\frac{c}{n-1}\int \frac{\sin cx}{x^{n-1}}dx+C\text{(za }n\ne 1\text{)}$

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

$\int \frac{dx}{{\cos }^{n}cx}=\frac{\sin cx}{c(n-1)co{s}^{n-1}cx}+\frac{n-2}{n-1}\int \frac{dx}{{\cos }^{n-2}cx}+C\text{(za }n>1\text{)}$

$\int \frac{dx}{1+\cos cx}=\frac{1}{c}\mathrm{tg}\frac{cx}{2}+C$

$\int \frac{dx}{1-\cos cx}=-\frac{1}{c}\mathrm{ctg}\frac{cx}{2}+C$

$\int \frac{xdx}{1+\cos cx}=\frac{x}{c}\mathrm{tg}\frac{cx}{2}+\frac{2}{c^2}\ln |\cos \frac{cx}{2}|+C$

$\int \frac{xdx}{1-\cos cx}=-\frac{x}{c}\mathrm{ctg}\frac{cx}{2}+\frac{2}{c^2}\ln |\sin \frac{cx}{2}|+C$

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

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

$\int \cos c_{1}x\cos c_{2}xdx=\frac{\sin (c_1-c_2)x}{2(c_1-c_2)}+\frac{\sin (c_1+c_2)x}{2(c_1+c_2)}+C\text{(za }|c_1|\ne |c_2|\text{)}$

Integrali trigonometrijskih funkcija koje sadrže samo tg

$\int \mathrm{tg}cxdx=-\frac{1}{c}\ln |\cos cx|=\frac{1}{c}\ln |\sec cx|+C$

$\int \frac{dx}{\mathrm{tg}cx}=\frac{1}{c}\ln |\sin cx|+C$

$\int {\mathrm{tg}}^{n}cxdx=\frac{1}{c(n-1)}{\mathrm{tg}}^{n-1}cx-\int {\mathrm{tg}}^{n-2}cxdx+C\text{(za }n\ne 1\text{)}$

$\int \frac{dx}{\mathrm{tg}cx+1}=\frac{x}{2}+\frac{1}{2c}\ln |\sin cx+\cos cx|+C$

$\int \frac{dx}{\mathrm{tg}cx-1}=-\frac{x}{2}+\frac{1}{2c}\ln |\sin cx-\cos cx|+C$

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

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

Integrali trigonometrijskih funkcija koje sadrže samo sec

$\sec cxdx=\frac{1}{c}\ln |\sec cx+\mathrm{tg}cx|+C$

${\sec }^{n}cxdx=\frac{{\sec }^{n-1}cx\sin cx}{c(n-1)}+\frac{n-2}{n-1}{\sec }^{n-2}cxdx+C\text{ (za }n\ne 1\text{)}$

$\frac{dx}{\sec x+1}=x-\mathrm{tg}\frac{x}{2}+C$

Integrali trigonometrijskih funkcija koje sadrže samo csc

$\int \csc cxdx=-\frac{1}{c}\ln |\csc cx+\mathrm{ctg}cx|+C$

$\int {\csc }^{2}xdx=-\mathrm{ctg}x+C$

$\int {\csc }^{n}cxdx=-\frac{{\csc }^{n-1}cx\cos cx}{c(n-1)}+\frac{n-2}{n-1}\int {\csc }^{n-2}cxdx+C\text{ (za }n\ne 1\text{)}$

Integrali trigonometrijskih funkcija koje sadrže samo ctg

$\int \mathrm{ctg}cxdx=\frac{1}{c}\ln |\sin cx|+C$

$\int {\mathrm{ctg}}^{n}cxdx=-\frac{1}{c(n-1)}{\mathrm{ctg}}^{n-1}cx-\int {\mathrm{ctg}}^{n-2}cxdx+C\text{(za }n\ne 1\text{)}$

$\int \frac{dx}{1+\mathrm{ctg}cx}=\int \frac{\mathrm{tg}cxdx}{\mathrm{tg}cx+1}+C$

$\int \frac{dx}{1-\mathrm{ctg}cx}=\int \frac{\mathrm{tg}cxdx}{\mathrm{tg}cx-1}+C$

Integrali trigonometrijskih funkcija koje sadrže i sin i cos

$\int \frac{dx}{\cos cx\pm \sin cx}=\frac{1}{c\sqrt{2}}\ln |\mathrm{tg}(\frac{cx}{2}\pm \frac{\pi }{8})|+C$

$\int \frac{dx}{(\cos cx\pm \sin cx{)}^2}=\frac{1}{2c}\mathrm{tg}(cx\mp \frac{\pi }{4})+C$

$\int \frac{dx}{(\cos x+\sin x{)}^n}=\frac{1}{n-1}(\frac{\sin x-\cos x}{(\cos x+\sin x{)}^{n-1}}-2(n-2)\int \frac{dx}{(\cos x+\sin x{)}^{n-2}})+C$

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

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

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

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

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

$\int \frac{\cos cxdx}{\sin cx(1+-\cos cx)}=-\frac{1}{4c}{\mathrm{ctg}}^2\frac{cx}{2}-\frac{1}{2c}\ln |\mathrm{tg}\frac{cx}{2}|+C$

$\int \frac{\sin cxdx}{\cos cx(1+\sin cx)}=\frac{1}{4c}{\mathrm{ctg}}^2(\frac{cx}{2}+\frac{\pi }{4})+\frac{1}{2c}\ln |\mathrm{tg}(\frac{cx}{2}+\frac{\pi }{4})|+C$

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

$\int \sin cx\cos cxdx=\frac{1}{2c}{\sin }^{2}cx+C$

$\int \sin c_{1}x\cos c_{2}xdx=-\frac{\cos (c_1+c_2)x}{2(c_1+c_2)}-\frac{\cos (c_1-c_2)x}{2(c_1-c_2)}+C\text{(za }|c_1|\ne |c_2|\text{)}$

$\int {\sin }^{n}cx\cos cxdx=\frac{1}{c(n+1)}{\sin }^{n+1}cx+C\text{(za }n\ne 1\text{)}$

$\int \sin cx{\cos }^{n}cxdx=-\frac{1}{c(n+1)}{\cos }^{n+1}cx+C\text{(za }n\ne 1\text{)}$

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

također: $\int {\sin }^{n}cx{\cos }^{m}cxdx=\frac{{\sin }^{n+1}cx{\cos }^{m-1}cx}{c(n+m)}+\frac{m-1}{n+m}\int {\sin }^{n}cx{\cos }^{m-2}cxdx+C\text{(za }m,n>0\text{)}$

$\int \frac{dx}{\sin cx\cos cx}=\frac{1}{c}\ln |\mathrm{tg}cx|+C$

$\int \frac{dx}{\sin cx{\cos }^{n}cx}=\frac{1}{c(n-1){\cos }^{n-1}cx}+\int \frac{dx}{\sin cx{\cos }^{n-2}cx}+C\text{(za }n\ne 1\text{)}$

$\int \frac{dx}{{\sin }^{n}cx\cos cx}=-\frac{1}{c(n-1){\sin }^{n-1}cx}+\int \frac{dx}{{\sin }^{n-2}cx\cos cx}+C\text{(za }n\ne 1\text{)}$

$\int \frac{\sin cxdx}{{\cos }^{n}cx}=\frac{1}{c(n-1){\cos }^{n-1}cx}+C\text{(za }n\ne 1\text{)}$

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

$\int \frac{{\sin }^{2}cxdx}{{\cos }^{n}cx}=\frac{\sin cx}{c(n-1){\cos }^{n-1}cx}-\frac{1}{n-1}\int \frac{dx}{{\cos }^{n-2}cx}+C\text{(za }n\ne 1\text{)}$

$\int \frac{{\sin }^{n}cxdx}{\cos cx}=-\frac{{\sin }^{n-1}cx}{c(n-1)}+\int \frac{{\sin }^{n-2}cxdx}{\cos cx}+C\text{(za }n\ne 1\text{)}$

$\int \frac{{\sin }^{n}cxdx}{{\cos }^{m}cx}=\frac{{\sin }^{n+1}cx}{c(m-1){\cos }^{m-1}cx}-\frac{n-m+2}{m-1}\int \frac{{\sin }^{n}cxdx}{{\cos }^{m-2}cx}+C\text{(za }m\ne 1\text{)}$

također: $\int \frac{{\sin }^{n}cxdx}{{\cos }^{m}cx}=-\frac{{\sin }^{n-1}cx}{c(n-m){\cos }^{m-1}cx}+\frac{n-1}{n-m}\int \frac{{\sin }^{n-2}cxdx}{{\cos }^{m}cx}+C\text{(za }m\ne n\text{)}$

također: $\int \frac{{\sin }^{n}cxdx}{{\cos }^{m}cx}=\frac{{\sin }^{n-1}cx}{c(m-1){\cos }^{m-1}cx}-\frac{n-1}{n-1}\int \frac{{\sin }^{n-1}cxdx}{{\cos }^{m-2}cx}+C\text{(za }m\ne 1\text{)}$

$\int \frac{\cos cxdx}{{\sin }^{n}cx}=-\frac{1}{c(n-1){\sin }^{n-1}cx}+C\text{(za }n\ne 1\text{)}$

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

$\int \frac{{\cos }^{2}cxdx}{{\sin }^{n}cx}=-\frac{1}{n-1}(\frac{\cos cx}{c{\sin }^{n-1}cx)}+\int \frac{dx}{{\sin }^{n-2}cx})+C\text{(za }n\ne 1\text{)}$

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

također: $\int \frac{{\cos }^{n}cxdx}{{\sin }^{m}cx}=\frac{{\cos }^{n-1}cx}{c(n-m){\sin }^{m-1}cx}+\frac{n-1}{n-m}\int \frac{co{s}^{n-2}cxdx}{{\sin }^{m}cx}+C\text{(za }m\ne n\text{)}$

također: $\int \frac{{\cos }^{n}cxdx}{{\sin }^{m}cx}=-\frac{{\cos }^{n-1}cx}{c(m-1){\sin }^{m-1}cx}-\frac{n-1}{m-1}\int \frac{co{s}^{n-2}cxdx}{{\sin }^{m-2}cx}+C\text{(za }m\ne 1\text{)}$

Integrali trigonometrijskih funkcija koje sadrže i sin i tg

$\int \sin cx\mathrm{tg}cxdx=\frac{1}{c}(\ln |\sec cx+\mathrm{tg}cx|-\sin cx)+C$

$\int \frac{{\mathrm{tg}}^{n}cxdx}{{\sin }^{2}cx}=\frac{1}{c(n-1)}{\mathrm{tg}}^{n-1}(cx)+C\text{(za }n\ne 1\text{)}$

Integrali trigonometrijskih funkcija koje sadrže i cos i tg

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

Integrali trigonometrijskih funkcija koje sadrže i sin i ctg

$\int \frac{{\mathrm{ctg}}^{n}cxdx}{{\sin }^{2}cx}=\frac{1}{c(n+1)}{\mathrm{ctg}}^{n+1}cx+C\text{(za }n\ne -1\text{)}$

Integrali trigonometrijskih funkcija koje sadrže i cos i ctg

$\int \frac{{\mathrm{ctg}}^{n}cxdx}{{\cos }^{2}cx}=\frac{1}{c(1-n)}{\mathrm{tg}}^{1-n}cx+C\text{(za }n\ne 1\text{)}$

Integrali trigonometrijskih funkcija koje sadrže i tg i ctg

$\int \frac{{\mathrm{tg}}^m(cx)}{{\mathrm{ctg}}^n(cx)}dx=\frac{1}{c(m+n-1)}{\mathrm{tg}}^{m+n-1}(cx)-\int \frac{{\mathrm{tg}}^{m-2}(cx)}{{\mathrm{ctg}}^n(cx)}dx+C\text{(za }m+n\ne 1\text{)}$

Integrali trigonometrijskih funkcija sa simetričnim granicama

$\underset{-c}{\overset{c}{\int }}\sin xdx=0$

$\underset{-c}{\overset{c}{\int }}\cos xdx=2\underset{0}{\overset{c}{\int }}\cos xdx=2\underset{-c}{\overset{0}{\int }}\cos xdx$

$\underset{-c}{\overset{c}{\int }}\mathrm{tg}xdx=0$