Seznam integrálů trigonometrických funkcí

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

Předpokládá se nenulová hodnota konstanty c.

Integrály obsahující sin

Kde c je konstanta:

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

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

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

kde $\mathrm{c}\mathrm{v}\mathrm{s}x=1-\sin x.$

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

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

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

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

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

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

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

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

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

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

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

$\int \sin c_{1}x\sin c_{2}x\mathrm{d}x=\frac{\sin (c_1-c_2)x}{2(c_1-c_2)}-\frac{\sin (c_1+c_2)x}{2(c_1+c_2)}\text{(pro }|c_1|\ne |c_2|\text{)}$

Integrály obsahující cos

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

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

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

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

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

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

$\int \frac{\cos cx}{x^n}\mathrm{d}x=-\frac{\cos cx}{(n-1)x^{n-1}}-\frac{c}{n-1}\int \frac{\sin cx}{x^{n-1}}\mathrm{d}x\text{(pro }n\ne 1\text{)}$

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

$\int \frac{\mathrm{d}x}{{\cos }^{n}cx}=\frac{\sin cx}{c(n-1)co{s}^{n-1}cx}+\frac{n-2}{n-1}\int \frac{\mathrm{d}x}{{\cos }^{n-2}cx}\text{(pro }n>1\text{)}$

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

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

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

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

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

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

$\int \cos c_{1}x\cos c_{2}x\mathrm{d}x=\frac{\sin (c_1-c_2)x}{2(c_1-c_2)}+\frac{\sin (c_1+c_2)x}{2(c_1+c_2)}\text{(pro }|c_1|\ne |c_2|\text{)}$

Integrály obsahující tg

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

$\int {\tan }^{n}cx\mathrm{d}x=\frac{1}{c(n-1)}{\tan }^{n-1}cx-\int {\tan }^{n-2}cx\mathrm{d}x\text{(pro }n\ne 1\text{)}$

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

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

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

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

Integrály obsahující sec

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

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

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

Integrály obsahující csc

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

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

Integrály obsahující cotg

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

$\int {\cot }^{n}cx\mathrm{d}x=-\frac{1}{c(n-1)}{\cot }^{n-1}cx-\int {\cot }^{n-2}cx\mathrm{d}x\text{(pro }n\ne 1\text{)}$

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

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

Integrály obsahující sin a cos

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

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

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

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

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

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

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

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

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

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

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

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

$\int \sin c_{1}x\cos c_{2}x\mathrm{d}x=-\frac{\cos (c_1+c_2)x}{2(c_1+c_2)}-\frac{\cos (c_1-c_2)x}{2(c_1-c_2)}\text{(pro }|c_1|\ne |c_2|\text{)}$

$\int {\sin }^{n}cx\cos cx\mathrm{d}x=\frac{1}{c(n+1)}{\sin }^{n+1}cx\text{(pro }n\ne 1\text{)}$

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

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

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

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

$\int \frac{\mathrm{d}x}{\sin cx{\cos }^{n}cx}=\frac{1}{c(n-1){\cos }^{n-1}cx}+\int \frac{\mathrm{d}x}{\sin cx{\cos }^{n-2}cx}\text{(pro }n\ne 1\text{)}$

$\int \frac{\mathrm{d}x}{{\sin }^{n}cx\cos cx}=-\frac{1}{c(n-1){\sin }^{n-1}cx}+\int \frac{\mathrm{d}x}{{\sin }^{n-2}cx\cos cx}\text{(pro }n\ne 1\text{)}$

$\int \frac{\sin cx\mathrm{d}x}{{\cos }^{n}cx}=\frac{1}{c(n-1){\cos }^{n-1}cx}\text{(pro }n\ne 1\text{)}$

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

$\int \frac{{\sin }^{2}cx\mathrm{d}x}{{\cos }^{n}cx}=\frac{\sin cx}{c(n-1){\cos }^{n-1}cx}-\frac{1}{n-1}\int \frac{\mathrm{d}x}{{\cos }^{n-2}cx}\text{(pro }n\ne 1\text{)}$

$\int \frac{{\sin }^{n}cx\mathrm{d}x}{\cos cx}=-\frac{{\sin }^{n-1}cx}{c(n-1)}+\int \frac{{\sin }^{n-2}cx\mathrm{d}x}{\cos cx}\text{(pro }n\ne 1\text{)}$

$\int \frac{{\sin }^{n}cx\mathrm{d}x}{{\cos }^{m}cx}=\frac{{\sin }^{n+1}cx}{c(m-1){\cos }^{m-1}cx}-\frac{n-m+2}{m-1}\int \frac{{\sin }^{n}cx\mathrm{d}x}{{\cos }^{m-2}cx}\text{(pro }m\ne 1\text{)}$

také: $\int \frac{{\sin }^{n}cx\mathrm{d}x}{{\cos }^{m}cx}=-\frac{{\sin }^{n-1}cx}{c(n-m){\cos }^{m-1}cx}+\frac{n-1}{n-m}\int \frac{{\sin }^{n-2}cx\mathrm{d}x}{{\cos }^{m}cx}\text{(pro }m\ne n\text{)}$

také: $\int \frac{{\sin }^{n}cx\mathrm{d}x}{{\cos }^{m}cx}=\frac{{\sin }^{n-1}cx}{c(m-1){\cos }^{m-1}cx}-\frac{n-1}{n-1}\int \frac{{\sin }^{n-1}cx\mathrm{d}x}{{\cos }^{m-2}cx}\text{(pro }m\ne 1\text{)}$

$\int \frac{\cos cx\mathrm{d}x}{{\sin }^{n}cx}=-\frac{1}{c(n-1){\sin }^{n-1}cx}\text{(pro }n\ne 1\text{)}$

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

$\int \frac{{\cos }^{2}cx\mathrm{d}x}{{\sin }^{n}cx}=-\frac{1}{n-1}(\frac{\cos cx}{c{\sin }^{n-1}cx)}+\int \frac{\mathrm{d}x}{{\sin }^{n-2}cx})\text{(pro }n\ne 1\text{)}$

$\int \frac{{\cos }^{n}cx\mathrm{d}x}{{\sin }^{m}cx}=-\frac{{\cos }^{n+1}cx}{c(m-1){\sin }^{m-1}cx}-\frac{n-m-2}{m-1}\int \frac{co{s}^{n}cx\mathrm{d}x}{{\sin }^{m-2}cx}\text{(pro }m\ne 1\text{)}$

také: $\int \frac{{\cos }^{n}cx\mathrm{d}x}{{\sin }^{m}cx}=\frac{{\cos }^{n-1}cx}{c(n-m){\sin }^{m-1}cx}+\frac{n-1}{n-m}\int \frac{co{s}^{n-2}cx\mathrm{d}x}{{\sin }^{m}cx}\text{(pro }m\ne n\text{)}$

také: $\int \frac{{\cos }^{n}cx\mathrm{d}x}{{\sin }^{m}cx}=-\frac{{\cos }^{n-1}cx}{c(m-1){\sin }^{m-1}cx}-\frac{n-1}{m-1}\int \frac{co{s}^{n-2}cx\mathrm{d}x}{{\sin }^{m-2}cx}\text{(pro }m\ne 1\text{)}$

Integrály obsahující sin a tg

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

$\int \frac{{\tan }^{n}cx\mathrm{d}x}{{\sin }^{2}cx}=\frac{1}{c(n-1)}{\tan }^{n-1}(cx)\text{(pro }n\ne 1\text{)}$

Integrály obsahující cos a tg

$\int \frac{{\tan }^{n}cx\mathrm{d}x}{{\cos }^{2}cx}=\frac{1}{c(n+1)}{\tan }^{n+1}cx\text{(pro }n\ne -1\text{)}$

Integrály obsahující sin a cotg

$\int \frac{{\cot }^{n}cx\mathrm{d}x}{{\sin }^{2}cx}=\frac{1}{c(n+1)}{\cot }^{n+1}cx\text{(pro }n\ne -1\text{)}$

Integrály obsahující cos a cotg

$\int \frac{{\cot }^{n}cx\mathrm{d}x}{{\cos }^{2}cx}=\frac{1}{c(1-n)}{\tan }^{1-n}cx\text{(pro }n\ne 1\text{)}$

Integrály obsahující tg a cotg

$\int \frac{{\tan }^m(cx)}{{\cot }^n(cx)}\mathrm{d}x=\frac{1}{c(m+n-1)}{\tan }^{m+n-1}(cx)-\int \frac{{\tan }^{m-2}(cx)}{{\cot }^n(cx)}\mathrm{d}x\text{(pro }m+n\ne 1\text{)}$