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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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
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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

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