Vidi kôd stranice Popis integrala trigonometrijskih funkcija

<!--'''Popis integrala trigonometrijskih funkcija'''-->Slijedi popis [[integral]]a ([[antiderivacija]] funkcija) [[trigonometrijska funkcija|trigonometrijskih funkcija]]. Za potpun popis integrala funkcija, pogledati [[tablica integrala]] i [[Popisi integrala|popis integrala]]. Vidjeti također: [[trigonometrijski integral]].

Za konstantu ''c'' se pretpostavlja da je različita od nule.

== Integrali trigonometrijskih funkcija koje sadrže samo [[sinus|sin]] ==

Pri čemu je ''c'' konstanta:

: <math>\int\sin cx\;dx = -\frac{1}{c}\cos cx\,\! + C</math>

:<math>\int\ |sin x|\,dx = -\cos x\,\! + C</math>

: <math>\int\sin^n {cx}\;dx = -\frac{\sin^{n-1} cx\cos cx}{nc} + \frac{n-1}{n}\int\sin^{n-2} cx\;dx + C \qquad\mbox{(za }n>0\mbox{)}\,\!</math>

: <math>\int\sin^2 {cx}\;dx = \frac{x}{2} - \frac{1}{4c} \sin 2cx \! + C</math>

: <math>\int\sqrt{1 - \sin{x}}\,dx = \int\sqrt{\operatorname{cvs}{x}}\,dx = 2 \frac{\cos{\frac{x}{2}} + \sin{\frac{x}{2}}}{\cos{\frac{x}{2}} - \sin{\frac{x}{2}}} \sqrt{\operatorname{cvs}{x}} = 2\sqrt{1 + \sin{x}} + C</math>

gdje je cvs{x} funkcija [[koversinus]].

: <math>\int x\sin cx\;dx = \frac{\sin cx}{c^2}-\frac{x\cos cx}{c}\,\! + C</math>

: <math>\int x^n\sin cx\;dx = -\frac{x^n}{c}\cos cx+\frac{n}{c}\int x^{n-1}\cos cx\;dx + C \qquad\mbox{(za }n>0\mbox{)}\,\!</math>

: <math>\int_{\frac{-a}{2}}^{\frac{a}{2}} x^2\sin^2 {\frac{n\pi x}{a}}\;dx = \frac{a^3(n^2\pi^2-6)}{24n^2\pi^2} \qquad\mbox{(za }n=2,4,6...\mbox{)}\,\!</math>

: <math>\int\frac{\sin cx}{x} dx = \sum_{i=0}^\infty (-1)^i\frac{(cx)^{2i+1}}{(2i+1)\cdot (2i+1)!}\,\! + C</math>

: <math>\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</math>

: <math>\int\frac{dx}{\sin cx} = \frac{1}{c}\ln \left|\operatorname{tg}\frac{cx}{2}\right| + C</math>

: <math>\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 \qquad\mbox{(za }n>1\mbox{)}\,\!</math>

: <math>\int\frac{dx}{1\pm\sin cx} = \frac{1}{c}\operatorname{tg}\left(\frac{cx}{2}\mp\frac{\pi}{4}\right) + C</math>

: <math>\int\frac{x\;dx}{1+\sin cx} = \frac{x}{c}\operatorname{tg}\left(\frac{cx}{2} - \frac{\pi}{4}\right)+\frac{2}{c^2}\ln\left|\cos\left(\frac{cx}{2}-\frac{\pi}{4}\right)\right| + C</math>

: <math>\int\frac{x\;dx}{1-\sin cx} = \frac{x}{c}\operatorname{ctg}\left(\frac{\pi}{4} - \frac{cx}{2}\right)+\frac{2}{c^2}\ln\left|\sin\left(\frac{\pi}{4}-\frac{cx}{2}\right)\right| + C</math>

: <math>\int\frac{\sin cx\;dx}{1\pm\sin cx} = \pm x+\frac{1}{c}\operatorname{tg}\left(\frac{\pi}{4}\mp\frac{cx}{2}\right) + C</math>

: <math>\int\sin c_1x\sin c_2x\;dx = \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 \qquad\mbox{(za }|c_1|\neq|c_2|\mbox{)}\,\!</math>

== Integrali trigonometrijskih funkcija koje sadrže samo [[kosinus|cos]] ==

: <math>\int\cos cx\;dx = \frac{1}{c}\sin cx\,\! + C</math>

: <math>\int\cos^n cx\;dx = \frac{\cos^{n-1} cx\sin cx}{nc} + \frac{n-1}{n}\int\cos^{n-2} cx\;dx + C \qquad\mbox{(za }n>0\mbox{)}\,\!</math>

: <math>\int x\cos cx\;dx = \frac{\cos cx}{c^2} + \frac{x\sin cx}{c}\,\! + C</math>

: <math>\int x^n\cos cx\;dx = \frac{x^n\sin cx}{c} - \frac{n}{c}\int x^{n-1}\sin cx\;dx\,\! + C</math>

: <math>\int_{\frac{-a}{2}}^{\frac{a}{2}} x^2\cos^2 {\frac{n\pi x}{a}}\;dx = \frac{a^3(n^2\pi^2-6)}{24n^2\pi^2}  \qquad\mbox{(za }n=1,3,5...\mbox{)}\,\!</math>

: <math>\int\frac{\cos cx}{x} dx = \ln|cx|+\sum_{i=1}^\infty (-1)^i\frac{(cx)^{2i}}{2i\cdot(2i)!}\,\! + C</math>

: <math>\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 \qquad\mbox{(za }n\neq 1\mbox{)}\,\!</math>

: <math>\int\frac{dx}{\cos cx} = \frac{1}{c}\ln\left|\operatorname{tg}\left(\frac{cx}{2}+\frac{\pi}{4}\right)\right| + C</math>

: <math>\int\frac{dx}{\cos^n cx} = \frac{\sin cx}{c(n-1) cos^{n-1} cx} + \frac{n-2}{n-1}\int\frac{dx}{\cos^{n-2} cx} + C \qquad\mbox{(za }n>1\mbox{)}\,\!</math>

: <math>\int\frac{dx}{1+\cos cx} = \frac{1}{c}\operatorname{tg}\frac{cx}{2}\,\! + C</math>

: <math>\int\frac{dx}{1-\cos cx} = -\frac{1}{c}\operatorname{ctg}\frac{cx}{2}\,\! + C</math>

: <math>\int\frac{x\;dx}{1+\cos cx} = \frac{x}{c}\operatorname{tg}\frac{cx}{2} + \frac{2}{c^2}\ln\left|\cos\frac{cx}{2}\right| + C</math>

: <math>\int\frac{x\;dx}{1-\cos cx} = -\frac{x}{c}\operatorname{ctg}\frac{cx}{2}+\frac{2}{c^2}\ln\left|\sin\frac{cx}{2}\right| + C</math>

: <math>\int\frac{\cos cx\;dx}{1+\cos cx} = x - \frac{1}{c}\operatorname{tg}\frac{cx}{2}\,\! + C</math>

: <math>\int\frac{\cos cx\;dx}{1-\cos cx} = -x-\frac{1}{c}\operatorname{ctg}\frac{cx}{2}\,\! + C</math>

: <math>\int\cos c_1x\cos c_2x\;dx = \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 \qquad\mbox{(za }|c_1|\neq|c_2|\mbox{)}\,\!</math>

== Integrali trigonometrijskih funkcija koje sadrže samo [[tangens|tg]] ==

: <math>\int\operatorname{tg} cx\;dx = -\frac{1}{c}\ln|\cos cx|\,\! = \frac{1}{c}\ln|\sec cx|\,\! + C</math>

: <math>\int\frac{dx}{\operatorname{tg} cx} = \frac{1}{c}\ln|\sin cx|\,\! + C</math>

: <math>\int\operatorname{tg}^n cx\;dx = \frac{1}{c(n-1)}\operatorname{tg}^{n-1} cx-\int\operatorname{tg}^{n-2} cx\;dx + C \qquad\mbox{(za }n\neq 1\mbox{)}\,\!</math>

: <math>\int\frac{dx}{\operatorname{tg} cx + 1} = \frac{x}{2} + \frac{1}{2c}\ln|\sin cx + \cos cx|\,\! + C</math>

: <math>\int\frac{dx}{\operatorname{tg} cx - 1} = -\frac{x}{2} + \frac{1}{2c}\ln|\sin cx - \cos cx|\,\! + C</math>

: <math>\int\frac{\operatorname{tg} cx\;dx}{\operatorname{tg} cx + 1} = \frac{x}{2} - \frac{1}{2c}\ln|\sin cx + \cos cx|\,\! + C</math>

: <math>\int\frac{\operatorname{tg} cx\;dx}{\operatorname{tg} cx - 1} = \frac{x}{2} + \frac{1}{2c}\ln|\sin cx - \cos cx|\,\! + C</math>

== Integrali trigonometrijskih funkcija koje sadrže samo [[sekans|sec]] ==

:<math>\ \sec{cx} \, dx = \frac{1}{c}\ln{\left| \sec{cx} + \operatorname{tg}{cx}\right|} + C</math>

:<math>\ \sec^n{cx} \, dx = \frac{\sec^{n-1}{cx} \sin {cx}}{c(n-1)} \,+\, \frac{n-2}{n-1}\ \sec^{n-2}{cx} \, dx + C \qquad \mbox{ (za }n \ne 1\mbox{)}\,\!</math>

:<math>\ \frac{dx}{\sec{x} + 1} = x - \operatorname{tg}{\frac{x}{2}} + C</math>

== Integrali trigonometrijskih funkcija koje sadrže samo [[kosekans|csc]] ==

:<math>\int \csc{cx} \, dx = -\frac{1}{c}\ln{\left| \csc{cx} + \operatorname{ctg}{cx}\right|} + C</math>

:<math>\int \csc^2{x} \, dx = -\operatorname{ctg}{x} + C</math>

:<math>\int \csc^n{cx} \, dx = -\frac{\csc^{n-1}{cx} \cos{cx}}{c(n-1)} \,+\, \frac{n-2}{n-1}\int \csc^{n-2}{cx} \, dx + C \qquad \mbox{ (za }n \ne 1\mbox{)}\,\!</math>

== Integrali trigonometrijskih funkcija koje sadrže samo [[kotangens|ctg]] ==

: <math>\int\operatorname{ctg} cx\;dx = \frac{1}{c}\ln|\sin cx|\,\! + C</math>

: <math>\int\operatorname{ctg}^n cx\;dx = -\frac{1}{c(n-1)}\operatorname{ctg}^{n-1} cx - \int\operatorname{ctg}^{n-2} cx\;dx + C \qquad\mbox{(za }n\neq 1\mbox{)}\,\!</math>

: <math>\int\frac{dx}{1 + \operatorname{ctg} cx} = \int\frac{\operatorname{tg} cx\;dx}{\operatorname{tg} cx+1}\,\! + C</math>

: <math>\int\frac{dx}{1 - \operatorname{ctg} cx} = \int\frac{\operatorname{tg} cx\;dx}{\operatorname{tg} cx-1}\,\! + C</math>

== Integrali trigonometrijskih funkcija koje sadrže i [[sinus|sin]] i [[kosinus|cos]] ==

: <math>\int\frac{dx}{\cos cx\pm\sin cx} = \frac{1}{c\sqrt{2}}\ln\left|\operatorname{tg}\left(\frac{cx}{2}\pm\frac{\pi}{8}\right)\right| + C</math>

: <math>\int\frac{dx}{(\cos cx\pm\sin cx)^2} = \frac{1}{2c}\operatorname{tg}\left(cx\mp\frac{\pi}{4}\right) + C</math>

: <math>\int\frac{dx}{(\cos x + \sin x)^n} = \frac{1}{n-1}\left(\frac{\sin x - \cos x}{(\cos x + \sin x)^{n - 1}} - 2(n - 2)\int\frac{dx}{(\cos x + \sin x)^{n-2}} \right) + C</math>

: <math>\int\frac{\cos cx\;dx}{\cos cx + \sin cx} = \frac{x}{2} + \frac{1}{2c}\ln\left|\sin cx + \cos cx\right| + C</math>

: <math>\int\frac{\cos cx\;dx}{\cos cx - \sin cx} = \frac{x}{2} - \frac{1}{2c}\ln\left|\sin cx - \cos cx\right| + C</math>

: <math>\int\frac{\sin cx\;dx}{\cos cx + \sin cx} = \frac{x}{2} - \frac{1}{2c}\ln\left|\sin cx + \cos cx\right| + C</math>

: <math>\int\frac{\sin cx\;dx}{\cos cx - \sin cx} = -\frac{x}{2} - \frac{1}{2c}\ln\left|\sin cx - \cos cx\right| + C</math>

: <math>\int\frac{\cos cx\;dx}{\sin cx(1+\cos cx)} = -\frac{1}{4c}\operatorname{tg}^2\frac{cx}{2}+\frac{1}{2c}\ln\left|\operatorname{tg}\frac{cx}{2}\right| + C</math>

: <math>\int\frac{\cos cx\;dx}{\sin cx(1+-\cos cx)} = -\frac{1}{4c}\operatorname{ctg}^2\frac{cx}{2}-\frac{1}{2c}\ln\left|\operatorname{tg}\frac{cx}{2}\right| + C</math>

: <math>\int\frac{\sin cx\;dx}{\cos cx(1+\sin cx)} = \frac{1}{4c}\operatorname{ctg}^2\left(\frac{cx}{2}+\frac{\pi}{4}\right)+\frac{1}{2c}\ln\left|\operatorname{tg}\left(\frac{cx}{2}+\frac{\pi}{4}\right)\right| + C</math>

: <math>\int\frac{\sin cx\;dx}{\cos cx(1-\sin cx)} = \frac{1}{4c}\operatorname{tg}^2\left(\frac{cx}{2}+\frac{\pi}{4}\right)-\frac{1}{2c}\ln\left|\operatorname{tg}\left(\frac{cx}{2}+\frac{\pi}{4}\right)\right| + C</math>

: <math>\int\sin cx\cos cx\;dx = \frac{1}{2c}\sin^2 cx\,\! + C</math>

: <math>\int\sin c_1x\cos c_2x\;dx = -\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 \qquad\mbox{(za }|c_1|\neq|c_2|\mbox{)}\,\!</math>

: <math>\int\sin^n cx\cos cx\;dx = \frac{1}{c(n+1)}\sin^{n+1} cx + C \qquad\mbox{(za }n\neq 1\mbox{)}\,\!</math>

: <math>\int\sin cx\cos^n cx\;dx = -\frac{1}{c(n+1)}\cos^{n+1} cx + C \qquad\mbox{(za }n\neq 1\mbox{)}\,\!</math>

: <math>\int\sin^n cx\cos^m cx\;dx = -\frac{\sin^{n-1} cx\cos^{m+1} cx}{c(n+m)}+\frac{n-1}{n+m}\int\sin^{n-2} cx\cos^m cx\;dx  + C \qquad\mbox{(za }m,n>0\mbox{)}\,\!</math>

: također: <math>\int\sin^n cx\cos^m cx\;dx = \frac{\sin^{n+1} cx\cos^{m-1} cx}{c(n+m)} + \frac{m-1}{n+m}\int\sin^n cx\cos^{m-2} cx\;dx + C \qquad\mbox{(za }m,n>0\mbox{)}\,\!</math>

: <math>\int\frac{dx}{\sin cx\cos cx} = \frac{1}{c}\ln\left|\operatorname{tg} cx\right| + C</math>

: <math>\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 \qquad\mbox{(za }n\neq 1\mbox{)}\,\!</math>

: <math>\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 \qquad\mbox{(za }n\neq 1\mbox{)}\,\!</math>

: <math>\int\frac{\sin cx\;dx}{\cos^n cx} = \frac{1}{c(n-1)\cos^{n-1} cx} + C \qquad\mbox{(za }n\neq 1\mbox{)}\,\!</math>

: <math>\int\frac{\sin^2 cx\;dx}{\cos cx} = -\frac{1}{c}\sin cx+\frac{1}{c}\ln\left|\operatorname{tg}\left(\frac{\pi}{4}+\frac{cx}{2}\right)\right| + C</math>

: <math>\int\frac{\sin^2 cx\;dx}{\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 \qquad\mbox{(za }n\neq 1\mbox{)}\,\!</math>

: <math>\int\frac{\sin^n cx\;dx}{\cos cx} = -\frac{\sin^{n-1} cx}{c(n-1)} + \int\frac{\sin^{n-2} cx\;dx}{\cos cx} + C \qquad\mbox{(za }n\neq 1\mbox{)}\,\!</math>

: <math>\int\frac{\sin^n cx\;dx}{\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 cx\;dx}{\cos^{m-2} cx} + C \qquad\mbox{(za }m\neq 1\mbox{)}\,\!</math>

: također: <math>\int\frac{\sin^n cx\;dx}{\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} cx\;dx}{\cos^m cx} + C \qquad\mbox{(za }m\neq n\mbox{)}\,\!</math>

: također: <math>\int\frac{\sin^n cx\;dx}{\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} cx\;dx}{\cos^{m-2} cx} + C \qquad\mbox{(za }m\neq 1\mbox{)}\,\!</math>

: <math>\int\frac{\cos cx\;dx}{\sin^n cx} = -\frac{1}{c(n-1)\sin^{n-1} cx} + C \qquad\mbox{(za }n\neq 1\mbox{)}\,\!</math>

: <math>\int\frac{\cos^2 cx\;dx}{\sin cx} = \frac{1}{c}\left(\cos cx+\ln\left|\operatorname{tg}\frac{cx}{2}\right|\right) + C</math>

: <math>\int\frac{\cos^2 cx\;dx}{\sin^n cx} = -\frac{1}{n-1}\left(\frac{\cos cx}{c\sin^{n-1} cx)}+\int\frac{dx}{\sin^{n-2} cx}\right) + C \qquad\mbox{(za }n\neq 1\mbox{)}</math>

: <math>\int\frac{\cos^n cx\;dx}{\sin^m cx} = -\frac{\cos^{n+1} cx}{c(m-1)\sin^{m-1} cx} - \frac{n-m-2}{m-1}\int\frac{cos^n cx\;dx}{\sin^{m-2} cx} + C \qquad\mbox{(za }m\neq 1\mbox{)}\,\!</math>

: također: <math>\int\frac{\cos^n cx\;dx}{\sin^m cx} = \frac{\cos^{n-1} cx}{c(n-m)\sin^{m-1} cx} + \frac{n-1}{n-m}\int\frac{cos^{n-2} cx\;dx}{\sin^m cx} + C \qquad\mbox{(za }m\neq n\mbox{)}\,\!</math>

: također: <math>\int\frac{\cos^n cx\;dx}{\sin^m cx} = -\frac{\cos^{n-1} cx}{c(m-1)\sin^{m-1} cx} - \frac{n-1}{m-1}\int\frac{cos^{n-2} cx\;dx}{\sin^{m-2} cx} + C \qquad\mbox{(za }m\neq 1\mbox{)}\,\!</math>

== Integrali trigonometrijskih funkcija koje sadrže i [[sinus|sin]] i [[tangens|tg]] ==

: <math>\int \sin cx \operatorname{tg} cx\;dx = \frac{1}{c}(\ln|\sec cx + \operatorname{tg} cx| - \sin cx)\,\! + C</math>

: <math>\int\frac{\operatorname{tg}^n cx\;dx}{\sin^2 cx} = \frac{1}{c(n-1)}\operatorname{tg}^{n-1} (cx) + C \qquad\mbox{(za }n\neq 1\mbox{)}\,\!</math>

== Integrali trigonometrijskih funkcija koje sadrže i [[kosinus|cos]] i [[tangens|tg]] ==

: <math>\int\frac{\operatorname{tg}^n cx\;dx}{\cos^2 cx} = \frac{1}{c(n+1)}\operatorname{tg}^{n+1} cx + C \qquad\mbox{(za }n\neq -1\mbox{)}\,\!</math>

== Integrali trigonometrijskih funkcija koje sadrže i [[sinus|sin]] i [[kotangens|ctg]] ==

: <math>\int\frac{\operatorname{ctg}^n cx\;dx}{\sin^2 cx} = \frac{1}{c(n+1)}\operatorname{ctg}^{n+1} cx  + C \qquad\mbox{(za }n\neq -1\mbox{)}\,\!</math>

== Integrali trigonometrijskih funkcija koje sadrže i [[kosinus|cos]] i [[kotangens|ctg]] ==

: <math>\int\frac{\operatorname{ctg}^n cx\;dx}{\cos^2 cx} = \frac{1}{c(1-n)}\operatorname{tg}^{1-n} cx + C \qquad\mbox{(za }n\neq 1\mbox{)}\,\!</math>

== Integrali trigonometrijskih funkcija koje sadrže i [[tangens|tg]] i [[kotangens|ctg]] ==

: <math>\int \frac{\operatorname{tg}^m(cx)}{\operatorname{ctg}^n(cx)}\;dx = \frac{1}{c(m+n-1)}\operatorname{tg}^{m+n-1}(cx) - \int \frac{\operatorname{tg}^{m-2}(cx)}{\operatorname{ctg}^n(cx)}\;dx+ C \qquad\mbox{(za }m + n \neq 1\mbox{)}\,\!</math>

== Integrali trigonometrijskih funkcija sa simetričnim granicama ==

: <math>\int_{{-c}}^{{c}}\sin {x}\;dx = 0</math>
: <math>\int_{{-c}}^{{c}}\cos {x}\;dx = 2\int_{{0}}^{{c}}\cos {x}\;dx = 2\int_{{-c}}^{{0}}\cos {x}\;dx \!</math>
: <math>\int_{{-c}}^{{c}}\operatorname{tg} {x}\;dx = 0</math>

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