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.

Pri čemu je c konstanta:

${\displaystyle \int \sin cx\;dx=-{\frac {1}{c}\cos cx\,\!+C}$

${\displaystyle \int \ |sinx|\,dx=-\cos x\,\!+C}$

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

${\displaystyle \int \sin ^{2}{cx}\;dx={\frac {x}{2}-{\frac {1}{4c}\sin 2cx\!+C}$

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

gdje je cvs{x} funkcija koversinus.

${\displaystyle \int x\sin cx\;dx={\frac {\sin cx}{c^{2}-{\frac {x\cos cx}{c}\,\!+C}$

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

${\displaystyle \int _{\frac {-a}{2}^{\frac {a}{2}x^{2}\sin ^{2}{\frac {n\pi x}{a}\;dx={\frac {a^{3}(n^{2}\pi ^{2}-6)}{24n^{2}\pi ^{2}\qquad {\mbox{(za }n=2,4,6...{\mbox{)}\,\!}$

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

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

${\displaystyle \int {\frac {dx}{\sin cx}={\frac {1}{c}\ln \left|\operatorname {tg} {\frac {cx}{2}\right|+C}$

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

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

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

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

${\displaystyle \int {\frac {\sin cx\;dx}{1\pm \sin cx}=\pm x+{\frac {1}{c}\operatorname {tg} \left({\frac {\pi }{4}\mp {\frac {cx}{2}\right)+C}$

${\displaystyle \int \sin c_{1}x\sin c_{2}x\;dx={\frac {\sin(c_{1}-c_{2})x}{2(c_{1}-c_{2})}-{\frac {\sin(c_{1}+c_{2})x}{2(c_{1}+c_{2})}+C\qquad {\mbox{(za }|c_{1}|\neq |c_{2}|{\mbox{)}\,\!}$

${\displaystyle \int \cos cx\;dx={\frac {1}{c}\sin cx\,\!+C}$

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

${\displaystyle \int x\cos cx\;dx={\frac {\cos cx}{c^{2}+{\frac {x\sin cx}{c}\,\!+C}$

${\displaystyle \int x^{n}\cos cx\;dx={\frac {x^{n}\sin cx}{c}-{\frac {n}{c}\int x^{n-1}\sin cx\;dx\,\!+C}$

${\displaystyle \int _{\frac {-a}{2}^{\frac {a}{2}x^{2}\cos ^{2}{\frac {n\pi x}{a}\;dx={\frac {a^{3}(n^{2}\pi ^{2}-6)}{24n^{2}\pi ^{2}\qquad {\mbox{(za }n=1,3,5...{\mbox{)}\,\!}$

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

${\displaystyle \int {\frac {\cos cx}{x^{n}dx=-{\frac {\cos cx}{(n-1)x^{n-1}-{\frac {c}{n-1}\int {\frac {\sin cx}{x^{n-1}dx+C\qquad {\mbox{(za }n\neq 1{\mbox{)}\,\!}$

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

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

${\displaystyle \int {\frac {dx}{1+\cos cx}={\frac {1}{c}\operatorname {tg} {\frac {cx}{2}\,\!+C}$

${\displaystyle \int {\frac {dx}{1-\cos cx}=-{\frac {1}{c}\operatorname {ctg} {\frac {cx}{2}\,\!+C}$

${\displaystyle \int {\frac {x\;dx}{1+\cos cx}={\frac {x}{c}\operatorname {tg} {\frac {cx}{2}+{\frac {2}{c^{2}\ln \left|\cos {\frac {cx}{2}\right|+C}$

${\displaystyle \int {\frac {x\;dx}{1-\cos cx}=-{\frac {x}{c}\operatorname {ctg} {\frac {cx}{2}+{\frac {2}{c^{2}\ln \left|\sin {\frac {cx}{2}\right|+C}$

${\displaystyle \int {\frac {\cos cx\;dx}{1+\cos cx}=x-{\frac {1}{c}\operatorname {tg} {\frac {cx}{2}\,\!+C}$

${\displaystyle \int {\frac {\cos cx\;dx}{1-\cos cx}=-x-{\frac {1}{c}\operatorname {ctg} {\frac {cx}{2}\,\!+C}$

${\displaystyle \int \cos c_{1}x\cos c_{2}x\;dx={\frac {\sin(c_{1}-c_{2})x}{2(c_{1}-c_{2})}+{\frac {\sin(c_{1}+c_{2})x}{2(c_{1}+c_{2})}+C\qquad {\mbox{(za }|c_{1}|\neq |c_{2}|{\mbox{)}\,\!}$

${\displaystyle \int \operatorname {tg} cx\;dx=-{\frac {1}{c}\ln |\cos cx|\,\!={\frac {1}{c}\ln |\sec cx|\,\!+C}$

${\displaystyle \int {\frac {dx}{\operatorname {tg} cx}={\frac {1}{c}\ln |\sin cx|\,\!+C}$

${\displaystyle \int \operatorname {tg} ^{n}cx\;dx={\frac {1}{c(n-1)}\operatorname {tg} ^{n-1}cx-\int \operatorname {tg} ^{n-2}cx\;dx+C\qquad {\mbox{(za }n\neq 1{\mbox{)}\,\!}$

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

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

${\displaystyle \int {\frac {\operatorname {tg} cx\;dx}{\operatorname {tg} cx+1}={\frac {x}{2}-{\frac {1}{2c}\ln |\sin cx+\cos cx|\,\!+C}$

${\displaystyle \int {\frac {\operatorname {tg} cx\;dx}{\operatorname {tg} cx-1}={\frac {x}{2}+{\frac {1}{2c}\ln |\sin cx-\cos cx|\,\!+C}$

${\displaystyle \ \sec {cx}\,dx={\frac {1}{c}\ln {\left|\sec {cx}+\operatorname {tg} {cx}\right|}+C}$

${\displaystyle \ \sec ^{n}{cx}\,dx={\frac {\sec ^{n-1}{cx}\sin {cx}{c(n-1)}\,+\,{\frac {n-2}{n-1}\ \sec ^{n-2}{cx}\,dx+C\qquad {\mbox{ (za }n\neq 1{\mbox{)}\,\!}$

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

${\displaystyle \int \csc {cx}\,dx=-{\frac {1}{c}\ln {\left|\csc {cx}+\operatorname {ctg} {cx}\right|}+C}$

${\displaystyle \int \csc ^{2}{x}\,dx=-\operatorname {ctg} {x}+C}$

${\displaystyle \int \csc ^{n}{cx}\,dx=-{\frac {\csc ^{n-1}{cx}\cos {cx}{c(n-1)}\,+\,{\frac {n-2}{n-1}\int \csc ^{n-2}{cx}\,dx+C\qquad {\mbox{ (za }n\neq 1{\mbox{)}\,\!}$

${\displaystyle \int \operatorname {ctg} cx\;dx={\frac {1}{c}\ln |\sin cx|\,\!+C}$

${\displaystyle \int \operatorname {ctg} ^{n}cx\;dx=-{\frac {1}{c(n-1)}\operatorname {ctg} ^{n-1}cx-\int \operatorname {ctg} ^{n-2}cx\;dx+C\qquad {\mbox{(za }n\neq 1{\mbox{)}\,\!}$

${\displaystyle \int {\frac {dx}{1+\operatorname {ctg} cx}=\int {\frac {\operatorname {tg} cx\;dx}{\operatorname {tg} cx+1}\,\!+C}$

${\displaystyle \int {\frac {dx}{1-\operatorname {ctg} cx}=\int {\frac {\operatorname {tg} cx\;dx}{\operatorname {tg} cx-1}\,\!+C}$

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

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

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

${\displaystyle \int {\frac {\cos cx\;dx}{\cos cx+\sin cx}={\frac {x}{2}+{\frac {1}{2c}\ln \left|\sin cx+\cos cx\right|+C}$

${\displaystyle \int {\frac {\cos cx\;dx}{\cos cx-\sin cx}={\frac {x}{2}-{\frac {1}{2c}\ln \left|\sin cx-\cos cx\right|+C}$

${\displaystyle \int {\frac {\sin cx\;dx}{\cos cx+\sin cx}={\frac {x}{2}-{\frac {1}{2c}\ln \left|\sin cx+\cos cx\right|+C}$

${\displaystyle \int {\frac {\sin cx\;dx}{\cos cx-\sin cx}=-{\frac {x}{2}-{\frac {1}{2c}\ln \left|\sin cx-\cos cx\right|+C}$

${\displaystyle \int {\frac {\cos cx\;dx}{\sin cx(1+\cos cx)}=-{\frac {1}{4c}\operatorname {tg} ^{2}{\frac {cx}{2}+{\frac {1}{2c}\ln \left|\operatorname {tg} {\frac {cx}{2}\right|+C}$

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

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

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

${\displaystyle \int \sin cx\cos cx\;dx={\frac {1}{2c}\sin ^{2}cx\,\!+C}$

${\displaystyle \int \sin c_{1}x\cos c_{2}x\;dx=-{\frac {\cos(c_{1}+c_{2})x}{2(c_{1}+c_{2})}-{\frac {\cos(c_{1}-c_{2})x}{2(c_{1}-c_{2})}+C\qquad {\mbox{(za }|c_{1}|\neq |c_{2}|{\mbox{)}\,\!}$

${\displaystyle \int \sin ^{n}cx\cos cx\;dx={\frac {1}{c(n+1)}\sin ^{n+1}cx+C\qquad {\mbox{(za }n\neq 1{\mbox{)}\,\!}$

${\displaystyle \int \sin cx\cos ^{n}cx\;dx=-{\frac {1}{c(n+1)}\cos ^{n+1}cx+C\qquad {\mbox{(za }n\neq 1{\mbox{)}\,\!}$

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

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

${\displaystyle \int {\frac {dx}{\sin cx\cos cx}={\frac {1}{c}\ln \left|\operatorname {tg} cx\right|+C}$

${\displaystyle \int {\frac {dx}{\sin cx\cos ^{n}cx}={\frac {1}{c(n-1)\cos ^{n-1}cx}+\int {\frac {dx}{\sin cx\cos ^{n-2}cx}+C\qquad {\mbox{(za }n\neq 1{\mbox{)}\,\!}$

${\displaystyle \int {\frac {dx}{\sin ^{n}cx\cos cx}=-{\frac {1}{c(n-1)\sin ^{n-1}cx}+\int {\frac {dx}{\sin ^{n-2}cx\cos cx}+C\qquad {\mbox{(za }n\neq 1{\mbox{)}\,\!}$

${\displaystyle \int {\frac {\sin cx\;dx}{\cos ^{n}cx}={\frac {1}{c(n-1)\cos ^{n-1}cx}+C\qquad {\mbox{(za }n\neq 1{\mbox{)}\,\!}$

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

${\displaystyle \int {\frac {\sin ^{2}cx\;dx}{\cos ^{n}cx}={\frac {\sin cx}{c(n-1)\cos ^{n-1}cx}-{\frac {1}{n-1}\int {\frac {dx}{\cos ^{n-2}cx}+C\qquad {\mbox{(za }n\neq 1{\mbox{)}\,\!}$

${\displaystyle \int {\frac {\sin ^{n}cx\;dx}{\cos cx}=-{\frac {\sin ^{n-1}cx}{c(n-1)}+\int {\frac {\sin ^{n-2}cx\;dx}{\cos cx}+C\qquad {\mbox{(za }n\neq 1{\mbox{)}\,\!}$

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

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

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

${\displaystyle \int {\frac {\cos cx\;dx}{\sin ^{n}cx}=-{\frac {1}{c(n-1)\sin ^{n-1}cx}+C\qquad {\mbox{(za }n\neq 1{\mbox{)}\,\!}$

${\displaystyle \int {\frac {\cos ^{2}cx\;dx}{\sin cx}={\frac {1}{c}\left(\cos cx+\ln \left|\operatorname {tg} {\frac {cx}{2}\right|\right)+C}$

${\displaystyle \int {\frac {\cos ^{2}cx\;dx}{\sin ^{n}cx}=-{\frac {1}{n-1}\left({\frac {\cos cx}{c\sin ^{n-1}cx)}+\int {\frac {dx}{\sin ^{n-2}cx}\right)+C\qquad {\mbox{(za }n\neq 1{\mbox{)}$

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

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

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

${\displaystyle \int \sin cx\operatorname {tg} cx\;dx={\frac {1}{c}(\ln |\sec cx+\operatorname {tg} cx|-\sin cx)\,\!+C}$

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

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

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

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

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

${\displaystyle \int _{-c}^{c}\sin {x}\;dx=0}$

${\displaystyle \int _{-c}^{c}\cos {x}\;dx=2\int _{0}^{c}\cos {x}\;dx=2\int _{-c}^{0}\cos {x}\;dx\!}$

${\displaystyle \int _{-c}^{c}\operatorname {tg} {x}\;dx=0}$

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