Tavola degli integrali indefiniti di funzioni trigonometriche

Questa pagina contiene una tavola di integrali indefiniti di funzioni trigonometriche.

Per altri integrali vedi Integrale § Tavole di integrali.

In questa pagina si assume che $c$ sia una costante diversa da 0.

Integrali di funzioni trigonometriche contenenti solo il seno

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Integrali di funzioni trigonometriche contenenti solo il coseno

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Integrali di funzioni trigonometriche contenenti solo tangente

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

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

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

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

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

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

Integrali di funzioni trigonometriche contenenti solo secante

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

$\int {\sec }^{n}cx\mathrm{d}x=\frac{{\sec }^{n-1}cx\sin cx}{c(n-1)}+\frac{n-2}{n-1}\int {\sec }^{n-2}cx\mathrm{d}x\text{per }n\ne 1,c\ne 0$

Integrali di funzioni trigonometriche contenenti solo cosecante

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

$\int {\csc }^{n}cx\mathrm{d}x=-\frac{{\csc }^{n-1}cx\cos cx}{c(n-1)}+\frac{n-2}{n-1}\int {\csc }^{n-2}cx\mathrm{d}x\text{per }n\ne 1,c\ne 0$

Integrali di funzioni trigonometriche contenenti solo cotangente

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

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

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

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

Integrali di funzioni trigonometriche contenenti seno e coseno

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Integrali di funzioni trigonometriche contenenti seno e tangente

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

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

Integrali di funzioni trigonometriche contenenti coseno e tangente

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

Integrali di funzioni trigonometriche contenenti seno e cotangente

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

Integrali di funzioni trigonometriche contenenti coseno e cotangente

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

Integrali di funzioni trigonometriche contenenti tangente e cotangente

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

Bibliografia

• Template:Cita libro