Seznam integrálů trigonometrických funkcí

Z Wikipedie, otevřené encyklopedie
Skočit na: Navigace, Hledání
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[editovat | editovat zdroj]

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[editovat | editovat zdroj]

\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[editovat | editovat zdroj]

\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[editovat | editovat zdroj]

\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[editovat | editovat zdroj]

\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[editovat | editovat zdroj]

\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[editovat | editovat zdroj]

\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[editovat | editovat zdroj]

\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[editovat | editovat zdroj]

\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[editovat | editovat zdroj]

\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[editovat | editovat zdroj]

\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[editovat | editovat zdroj]

\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{)}\,\!