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 ${\displaystyle c}$ sia una costante diversa da 0.

Lo stesso argomento in dettaglio: Seno (trigonometria).

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

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

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

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

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

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

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

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

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

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

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

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

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

${\displaystyle \int \sin c_{1}x\sin c_{2}x\;\mathrm {d} x={\frac {\sin(c_{1}-c_{2})x}{2(c_{1}-c_{2})}-{\frac {\sin(c_{1}+c_{2})x}{2(c_{1}+c_{2})}\qquad ({\text{per }|c_{1}|\neq |c_{2}|)}$

Lo stesso argomento in dettaglio: Coseno.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

${\displaystyle \int \cos c_{1}x\cos c_{2}x\;\mathrm {d} x={\frac {\sin(c_{1}-c_{2})x}{2(c_{1}-c_{2})}+{\frac {\sin(c_{1}+c_{2})x}{2(c_{1}+c_{2})}\qquad ({\text{per }|c_{1}|\neq |c_{2}|)}$

Lo stesso argomento in dettaglio: Tangente (matematica).

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

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

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

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

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

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

Lo stesso argomento in dettaglio: Secante (trigonometria).

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

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

Lo stesso argomento in dettaglio: Cosecante.

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

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

Lo stesso argomento in dettaglio: Cotangente.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

${\displaystyle \int \sin c_{1}x\cos c_{2}x\;\mathrm {d} x=-{\frac {\cos(c_{1}+c_{2})x}{2(c_{1}+c_{2})}-{\frac {\cos(c_{1}-c_{2})x}{2(c_{1}-c_{2})}\qquad ({\text{per }|c_{1}|\neq |c_{2}|)}$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

• Murray R. Spiegel, Manuale di matematica, Etas Libri, 1974, pp. 75-82.

V · D · M Trigonometria
Funzione trigonometrica · Funzione trigonometrica inversa
Funzioni	Seno · Senoverso · Coseno · Cosenoverso · Tangente · Cotangente · Secante · Secante esterna · Cosecante · Cosecante esterna · Funzioni trigonometriche complesse
Funzioni inverse	Arcoseno · Arcocoseno · Arcotangente · Arcocotangente · Arcosecante · Arcocosecante
Teoremi e formule	Teorema della corda · Teorema dei seni · Teorema del coseno · Teorema di Nepero · Teorema delle cotangenti · Teorema di Pitagora · Formule di prostaferesi · Formule di Werner · Formula di Eulero · Formule di duplicazione
Altro	Tavola trigonometrica · Tavola degli integrali indefiniti di funzioni trigonometriche · Tavola degli integrali indefiniti di funzioni d'arco · Funzioni iperboliche · Identità trigonometrica · Costanti trigonometriche esatte · Storia delle funzioni trigonometriche · Equazione trigonometrica · Disequazione trigonometrica

Portale Matematica: accedi alle voci di Wikipedia che trattano di matematica