Seznam integrálů trigonometrických funkcí

Seznamy integrálů

Tabulka integrálů elementárních funkcí
racionální funkce
iracionální funkce
exponenciální funkce
logaritmické funkce
trigonometrické funkce
inverzní trigonometrické funkce
hyperbolické funkce
inverzní hyperbolické funkce

Toto je seznam integrálů (primitivních funkcí) pro integrandy obsahující trigonometrické funkce.

Předpokládá se nenulová hodnota konstanty c.

Integrály obsahující sin

Kde c je konstanta:

$\int\sin cx\;\mathrm{d}x = -\frac{1}{c}\cos cx\,\!$

$\int\sin^n {cx}\;\mathrm{d}x = -\frac{\sin^{n-1} cx\cos cx}{nc} + \frac{n-1}{n}\int\sin^{n-2} cx\;\mathrm{d}x \qquad\mbox{(pro }n>0\mbox{)}\,\!$

$\int\sqrt{1 - \sin{x}}\,\mathrm{d}x = \int\sqrt{\operatorname{cvs}\,{x}}\,\mathrm{d}x = 2 \frac{\cos{\frac{x}{2}} + \sin{\frac{x}{2}}}{\cos{\frac{x}{2}} - \sin{\frac{x}{2}}} \sqrt{\operatorname{cvs}\,{x}} = 2\sqrt{1 + \sin{x}}$

kde $\mathrm{cvs}\,x = 1 - \sin x.$

$\int x\sin cx\;\mathrm{d}x = \frac{\sin cx}{c^2}-\frac{x\cos cx}{c}\,\!$

$\int x^n\sin cx\;\mathrm{d}x = -\frac{x^n}{c}\cos cx+\frac{n}{c}\int x^{n-1}\cos cx\;\mathrm{d}x \qquad\mbox{(pro }n>0\mbox{)}\,\!$

$\int_{\frac{-a}{2}}^{\frac{a}{2}} x^2\sin^2 {\frac{n\pi x}{a}}\;\mathrm{d}x = \frac{a^3(n^2\pi^2-6)}{24n^2\pi^2} \qquad\mbox{(pro }n=2,4,6...\mbox{)}\,\!$

$\int\frac{\sin cx}{x} \mathrm{d}x = \sum_{i=0}^\infty (-1)^i\frac{(cx)^{2i+1}}{(2i+1)\cdot (2i+1)!}\,\!$

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

$\int\frac{\mathrm{d}x}{\sin cx} = \frac{1}{c}\ln \left|\tan\frac{cx}{2}\right|$

$\int\frac{\mathrm{d}x}{\sin^n cx} = \frac{\cos cx}{c(1-n) \sin^{n-1} cx}+\frac{n-2}{n-1}\int\frac{\mathrm{d}x}{\sin^{n-2}cx} \qquad\mbox{(pro }n>1\mbox{)}\,\!$

$\int\frac{\mathrm{d}x}{1\pm\sin cx} = \frac{1}{c}\tan\left(\frac{cx}{2}\mp\frac{\pi}{4}\right)$

$\int\frac{x\;\mathrm{d}x}{1+\sin cx} = \frac{x}{c}\tan\left(\frac{cx}{2} - \frac{\pi}{4}\right)+\frac{2}{c^2}\ln\left|\cos\left(\frac{cx}{2}-\frac{\pi}{4}\right)\right|$

$\int\frac{x\;\mathrm{d}x}{1-\sin cx} = \frac{x}{c}\cot\left(\frac{\pi}{4} - \frac{cx}{2}\right)+\frac{2}{c^2}\ln\left|\sin\left(\frac{\pi}{4}-\frac{cx}{2}\right)\right|$

$\int\frac{\sin cx\;\mathrm{d}x}{1\pm\sin cx} = \pm x+\frac{1}{c}\tan\left(\frac{\pi}{4}\mp\frac{cx}{2}\right)$

$\int\sin c_1x\sin c_2x\;\mathrm{d}x = \frac{\sin(c_1-c_2)x}{2(c_1-c_2)}-\frac{\sin(c_1+c_2)x}{2(c_1+c_2)} \qquad\mbox{(pro }|c_1|\neq|c_2|\mbox{)}\,\!$

Integrály obsahující cos

$\int\cos cx\;\mathrm{d}x = \frac{1}{c}\sin cx\,\!$

$\int\cos^n cx\;\mathrm{d}x = \frac{\cos^{n-1} cx\sin cx}{nc} + \frac{n-1}{n}\int\cos^{n-2} cx\;\mathrm{d}x \qquad\mbox{(pro }n>0\mbox{)}\,\!$

$\int x\cos cx\;\mathrm{d}x = \frac{\cos cx}{c^2} + \frac{x\sin cx}{c}\,\!$

$\int x^n\cos cx\;\mathrm{d}x = \frac{x^n\sin cx}{c} - \frac{n}{c}\int x^{n-1}\sin cx\;\mathrm{d}x\,\!$

$\int_{\frac{-a}{2}}^{\frac{a}{2}} x^2\cos^2 {\frac{n\pi x}{a}}\;\mathrm{d}x = \frac{a^3(n^2\pi^2-6)}{24n^2\pi^2} \qquad\mbox{(pro }n=1,3,5...\mbox{)}\,\!$

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

$\int\frac{\cos cx}{x^n} \mathrm{d}x = -\frac{\cos cx}{(n-1)x^{n-1}}-\frac{c}{n-1}\int\frac{\sin cx}{x^{n-1}} \mathrm{d}x \qquad\mbox{(pro }n\neq 1\mbox{)}\,\!$

$\int\frac{\mathrm{d}x}{\cos cx} = \frac{1}{c}\ln\left|\tan\left(\frac{cx}{2}+\frac{\pi}{4}\right)\right|$

$\int\frac{\mathrm{d}x}{\cos^n cx} = \frac{\sin cx}{c(n-1) cos^{n-1} cx} + \frac{n-2}{n-1}\int\frac{\mathrm{d}x}{\cos^{n-2} cx} \qquad\mbox{(pro }n>1\mbox{)}\,\!$

$\int\frac{\mathrm{d}x}{1+\cos cx} = \frac{1}{c}\tan\frac{cx}{2}\,\!$

$\int\frac{\mathrm{d}x}{1-\cos cx} = -\frac{1}{c}\cot\frac{cx}{2}\,\!$

$\int\frac{x\;\mathrm{d}x}{1+\cos cx} = \frac{x}{c}\tan\frac{cx}{2} + \frac{2}{c^2}\ln\left|\cos\frac{cx}{2}\right|$

$\int\frac{x\;\mathrm{d}x}{1-\cos cx} = -\frac{x}{c}\cot\frac{cx}{2}+\frac{2}{c^2}\ln\left|\sin\frac{cx}{2}\right|$

$\int\frac{\cos cx\;\mathrm{d}x}{1+\cos cx} = x - \frac{1}{c}\tan\frac{cx}{2}\,\!$

$\int\frac{\cos cx\;\mathrm{d}x}{1-\cos cx} = -x-\frac{1}{c}\cot\frac{cx}{2}\,\!$

$\int\cos c_1x\cos c_2x\;\mathrm{d}x = \frac{\sin(c_1-c_2)x}{2(c_1-c_2)}+\frac{\sin(c_1+c_2)x}{2(c_1+c_2)} \qquad\mbox{(pro }|c_1|\neq|c_2|\mbox{)}\,\!$

Integrály obsahující tg

$\int\tan cx\;\mathrm{d}x = -\frac{1}{c}\ln|\cos cx|\,\! = \frac{1}{c}\ln|\sec cx|\,\!$

$\int\tan^n cx\;\mathrm{d}x = \frac{1}{c(n-1)}\tan^{n-1} cx-\int\tan^{n-2} cx\;\mathrm{d}x \qquad\mbox{(pro }n\neq 1\mbox{)}\,\!$

$\int\frac{\mathrm{d}x}{\tan cx + 1} = \frac{x}{2} + \frac{1}{2c}\ln|\sin cx + \cos cx|\,\!$

$\int\frac{\mathrm{d}x}{\tan cx - 1} = -\frac{x}{2} + \frac{1}{2c}\ln|\sin cx - \cos cx|\,\!$

$\int\frac{\tan cx\;\mathrm{d}x}{\tan cx + 1} = \frac{x}{2} - \frac{1}{2c}\ln|\sin cx + \cos cx|\,\!$

$\int\frac{\tan cx\;\mathrm{d}x}{\tan cx - 1} = \frac{x}{2} + \frac{1}{2c}\ln|\sin cx - \cos cx|\,\!$

Integrály obsahující sec

$\int \sec{cx} \, \mathrm{d}x = \frac{1}{c}\ln{\left| \sec{cx} + \tan{cx}\right|}$

$\int \sec^n{cx} \, \mathrm{d}x = \frac{\sec^{n-1}{cx} \sin {cx}}{c(n-1)} \,+\, \frac{n-2}{n-1}\int \sec^{n-2}{cx} \, \mathrm{d}x \qquad \mbox{ (pro }n \ne 1\mbox{)}\,\!$

$\int \frac{\mathrm{d}x}{\sec{x} + 1} = x - \tan{\frac{x}{2}}$

Integrály obsahující csc

$\int \csc{cx} \, \mathrm{d}x = -\frac{1}{c}\ln{\left| \csc{cx} + \cot{cx}\right|}$

$\int \csc^n{cx} \, \mathrm{d}x = -\frac{\csc^{n-1}{cx} \cos{cx}}{c(n-1)} \,+\, \frac{n-2}{n-1}\int \csc^{n-2}{cx} \, \mathrm{d}x \qquad \mbox{ (pro }n \ne 1\mbox{)}\,\!$

Integrály obsahující cotg

$\int\cot cx\;\mathrm{d}x = \frac{1}{c}\ln|\sin cx|\,\!$

$\int\cot^n cx\;\mathrm{d}x = -\frac{1}{c(n-1)}\cot^{n-1} cx - \int\cot^{n-2} cx\;\mathrm{d}x \qquad\mbox{(pro }n\neq 1\mbox{)}\,\!$

$\int\frac{\mathrm{d}x}{1 + \cot cx} = \int\frac{\tan cx\;\mathrm{d}x}{\tan cx+1}\,\!$

$\int\frac{\mathrm{d}x}{1 - \cot cx} = \int\frac{\tan cx\;\mathrm{d}x}{\tan cx-1}\,\!$

Integrály obsahující sin a cos

$\int\frac{\mathrm{d}x}{\cos cx\pm\sin cx} = \frac{1}{c\sqrt{2}}\ln\left|\tan\left(\frac{cx}{2}\pm\frac{\pi}{8}\right)\right|$

$\int\frac{\mathrm{d}x}{(\cos cx\pm\sin cx)^2} = \frac{1}{2c}\tan\left(cx\mp\frac{\pi}{4}\right)$

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

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

$\int\frac{\cos cx\;\mathrm{d}x}{\cos cx - \sin cx} = \frac{x}{2} - \frac{1}{2c}\ln\left|\sin cx - \cos cx\right|$

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

$\int\frac{\sin cx\;\mathrm{d}x}{\cos cx - \sin cx} = -\frac{x}{2} - \frac{1}{2c}\ln\left|\sin cx - \cos cx\right|$

$\int\frac{\cos cx\;\mathrm{d}x}{\sin cx(1+\cos cx)} = -\frac{1}{4c}\tan^2\frac{cx}{2}+\frac{1}{2c}\ln\left|\tan\frac{cx}{2}\right|$

$\int\frac{\cos cx\;\mathrm{d}x}{\sin cx(1+-\cos cx)} = -\frac{1}{4c}\cot^2\frac{cx}{2}-\frac{1}{2c}\ln\left|\tan\frac{cx}{2}\right|$

$\int\frac{\sin cx\;\mathrm{d}x}{\cos cx(1+\sin cx)} = \frac{1}{4c}\cot^2\left(\frac{cx}{2}+\frac{\pi}{4}\right)+\frac{1}{2c}\ln\left|\tan\left(\frac{cx}{2}+\frac{\pi}{4}\right)\right|$

$\int\frac{\sin cx\;\mathrm{d}x}{\cos cx(1-\sin cx)} = \frac{1}{4c}\tan^2\left(\frac{cx}{2}+\frac{\pi}{4}\right)-\frac{1}{2c}\ln\left|\tan\left(\frac{cx}{2}+\frac{\pi}{4}\right)\right|$

$\int\sin cx\cos cx\;\mathrm{d}x = \frac{1}{2c}\sin^2 cx\,\!$

$\int\sin c_1x\cos c_2x\;\mathrm{d}x = -\frac{\cos(c_1+c_2)x}{2(c_1+c_2)}-\frac{\cos(c_1-c_2)x}{2(c_1-c_2)} \qquad\mbox{(pro }|c_1|\neq|c_2|\mbox{)}\,\!$

$\int\sin^n cx\cos cx\;\mathrm{d}x = \frac{1}{c(n+1)}\sin^{n+1} cx \qquad\mbox{(pro }n\neq 1\mbox{)}\,\!$

$\int\sin cx\cos^n cx\;\mathrm{d}x = -\frac{1}{c(n+1)}\cos^{n+1} cx \qquad\mbox{(pro }n\neq 1\mbox{)}\,\!$

$\int\sin^n cx\cos^m cx\;\mathrm{d}x = -\frac{\sin^{n-1} cx\cos^{m+1} cx}{c(n+m)}+\frac{n-1}{n+m}\int\sin^{n-2} cx\cos^m cx\;\mathrm{d}x \qquad\mbox{(pro }m,n>0\mbox{)}\,\!$

také: $\int\sin^n cx\cos^m cx\;\mathrm{d}x = \frac{\sin^{n+1} cx\cos^{m-1} cx}{c(n+m)} + \frac{m-1}{n+m}\int\sin^n cx\cos^{m-2} cx\;\mathrm{d}x \qquad\mbox{(pro }m,n>0\mbox{)}\,\!$

$\int\frac{\mathrm{d}x}{\sin cx\cos cx} = \frac{1}{c}\ln\left|\tan cx\right|$

$\int\frac{\mathrm{d}x}{\sin cx\cos^n cx} = \frac{1}{c(n-1)\cos^{n-1} cx}+\int\frac{\mathrm{d}x}{\sin cx\cos^{n-2} cx} \qquad\mbox{(pro }n\neq 1\mbox{)}\,\!$

$\int\frac{\mathrm{d}x}{\sin^n cx\cos cx} = -\frac{1}{c(n-1)\sin^{n-1} cx}+\int\frac{\mathrm{d}x}{\sin^{n-2} cx\cos cx} \qquad\mbox{(pro }n\neq 1\mbox{)}\,\!$

$\int\frac{\sin cx\;\mathrm{d}x}{\cos^n cx} = \frac{1}{c(n-1)\cos^{n-1} cx} \qquad\mbox{(pro }n\neq 1\mbox{)}\,\!$

$\int\frac{\sin^2 cx\;\mathrm{d}x}{\cos cx} = -\frac{1}{c}\sin cx+\frac{1}{c}\ln\left|\tan\left(\frac{\pi}{4}+\frac{cx}{2}\right)\right|$

$\int\frac{\sin^2 cx\;\mathrm{d}x}{\cos^n cx} = \frac{\sin cx}{c(n-1)\cos^{n-1}cx}-\frac{1}{n-1}\int\frac{\mathrm{d}x}{\cos^{n-2}cx} \qquad\mbox{(pro }n\neq 1\mbox{)}\,\!$

$\int\frac{\sin^n cx\;\mathrm{d}x}{\cos cx} = -\frac{\sin^{n-1} cx}{c(n-1)} + \int\frac{\sin^{n-2} cx\;\mathrm{d}x}{\cos cx} \qquad\mbox{(pro }n\neq 1\mbox{)}\,\!$

$\int\frac{\sin^n cx\;\mathrm{d}x}{\cos^m cx} = \frac{\sin^{n+1} cx}{c(m-1)\cos^{m-1} cx}-\frac{n-m+2}{m-1}\int\frac{\sin^n cx\;\mathrm{d}x}{\cos^{m-2} cx} \qquad\mbox{(pro }m\neq 1\mbox{)}\,\!$

také: $\int\frac{\sin^n cx\;\mathrm{d}x}{\cos^m cx} = -\frac{\sin^{n-1} cx}{c(n-m)\cos^{m-1} cx}+\frac{n-1}{n-m}\int\frac{\sin^{n-2} cx\;\mathrm{d}x}{\cos^m cx} \qquad\mbox{(pro }m\neq n\mbox{)}\,\!$

také: $\int\frac{\sin^n cx\;\mathrm{d}x}{\cos^m cx} = \frac{\sin^{n-1} cx}{c(m-1)\cos^{m-1} cx}-\frac{n-1}{n-1}\int\frac{\sin^{n-1} cx\;\mathrm{d}x}{\cos^{m-2} cx} \qquad\mbox{(pro }m\neq 1\mbox{)}\,\!$

$\int\frac{\cos cx\;\mathrm{d}x}{\sin^n cx} = -\frac{1}{c(n-1)\sin^{n-1} cx} \qquad\mbox{(pro }n\neq 1\mbox{)}\,\!$

$\int\frac{\cos^2 cx\;\mathrm{d}x}{\sin cx} = \frac{1}{c}\left(\cos cx+\ln\left|\tan\frac{cx}{2}\right|\right)$

$\int\frac{\cos^2 cx\;\mathrm{d}x}{\sin^n cx} = -\frac{1}{n-1}\left(\frac{\cos cx}{c\sin^{n-1} cx)}+\int\frac{\mathrm{d}x}{\sin^{n-2} cx}\right) \qquad\mbox{(pro }n\neq 1\mbox{)}$

$\int\frac{\cos^n cx\;\mathrm{d}x}{\sin^m cx} = -\frac{\cos^{n+1} cx}{c(m-1)\sin^{m-1} cx} - \frac{n-m-2}{m-1}\int\frac{cos^n cx\;\mathrm{d}x}{\sin^{m-2} cx} \qquad\mbox{(pro }m\neq 1\mbox{)}\,\!$

také: $\int\frac{\cos^n cx\;\mathrm{d}x}{\sin^m cx} = \frac{\cos^{n-1} cx}{c(n-m)\sin^{m-1} cx} + \frac{n-1}{n-m}\int\frac{cos^{n-2} cx\;\mathrm{d}x}{\sin^m cx} \qquad\mbox{(pro }m\neq n\mbox{)}\,\!$

také: $\int\frac{\cos^n cx\;\mathrm{d}x}{\sin^m cx} = -\frac{\cos^{n-1} cx}{c(m-1)\sin^{m-1} cx} - \frac{n-1}{m-1}\int\frac{cos^{n-2} cx\;\mathrm{d}x}{\sin^{m-2} cx} \qquad\mbox{(pro }m\neq 1\mbox{)}\,\!$

Integrály obsahující sin a tg

$\int \sin cx \tan cx\;\mathrm{d}x = \frac{1}{c}(\ln|\sec cx + \tan cx| - \sin cx)\,\!$

$\int\frac{\tan^n cx\;\mathrm{d}x}{\sin^2 cx} = \frac{1}{c(n-1)}\tan^{n-1} (cx) \qquad\mbox{(pro }n\neq 1\mbox{)}\,\!$

Integrály obsahující cos a tg

$\int\frac{\tan^n cx\;\mathrm{d}x}{\cos^2 cx} = \frac{1}{c(n+1)}\tan^{n+1} cx \qquad\mbox{(pro }n\neq -1\mbox{)}\,\!$

Integrály obsahující sin a cotg

$\int\frac{\cot^n cx\;\mathrm{d}x}{\sin^2 cx} = \frac{1}{c(n+1)}\cot^{n+1} cx \qquad\mbox{(pro }n\neq -1\mbox{)}\,\!$

Integrály obsahující cos a cotg

$\int\frac{\cot^n cx\;\mathrm{d}x}{\cos^2 cx} = \frac{1}{c(1-n)}\tan^{1-n} cx \qquad\mbox{(pro }n\neq 1\mbox{)}\,\!$

Integrály obsahující tg a cotg

$\int \frac{\tan^m(cx)}{\cot^n(cx)}\;\mathrm{d}x = \frac{1}{c(m+n-1)}\tan^{m+n-1}(cx) - \int \frac{\tan^{m-2}(cx)}{\cot^n(cx)}\;\mathrm{d}x\qquad\mbox{(pro }m + n \neq 1\mbox{)}\,\!$