Tavola degli integrali indefiniti di funzioni trigonometriche

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

Per altri integrali vedi Indici per la matematica#Tavole di integrali.

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

Integrali di funzioni trigonometriche contenenti solo il seno

$\int \mathrm {sen} \,cx\;dx=-{\frac {1}{c}}\cos cx$

$\int \mathrm {sen} ^{n}cx\;dx=-{\frac {\mathrm {sen} ^{n-1}cx\cos cx}{nc}}+{\frac {n-1}{n}}\int \mathrm {sen} ^{n-2}cx\;dx\qquad {\mbox{(per }}n>0{\mbox{)}}$

$\int x\mathrm {sen} \,cx\;dx={\frac {\mathrm {sen} \,cx}{c^{2}}}-{\frac {x\cos cx}{c}}$

$\int x^{n}\mathrm {sen} \,cx\;dx=-{\frac {x^{n}}{c}}\cos cx+{\frac {n}{c}}\int x^{n-1}\cos cx\;dx\qquad {\mbox{(per }}n>0{\mbox{)}}$

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

$\int {\frac {\mathrm {sen} \,cx}{x^{n}}}dx=-{\frac {\mathrm {sen} \,cx}{(n-1)x^{n-1}}}+{\frac {c}{n-1}}\int {\frac {\cos cx}{x^{n-1}}}dx$

$\int {\frac {dx}{\mathrm {sen} \,cx}}={\frac {1}{c}}\ln \left|\tan {\frac {cx}{2}}\right|$

$\int {\frac {dx}{\mathrm {sen} ^{n}cx}}={\frac {\cos cx}{c(1-n)\mathrm {sen} ^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\mathrm {sen} ^{n-2}cx}}\qquad {\mbox{(per }}n>1{\mbox{)}}$

$\int {\frac {dx}{1\pm \mathrm {sen} \,cx}}={\frac {1}{c}}\tan \left({\frac {cx}{2}}\mp {\frac {\pi }{4}}\right)$

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

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

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

$\int \mathrm {sen} \,c_{1}x\mathrm {sen} \,c_{2}x\;dx={\frac {\mathrm {sen} (c_{1}-c_{2})x}{2(c_{1}-c_{2})}}-{\frac {\mathrm {sen} (c_{1}+c_{2})x}{2(c_{1}+c_{2})}}\qquad {\mbox{(per }}|c_{1}|\neq |c_{2}|{\mbox{)}}$

Integrali di funzioni trigonometriche contenenti solo il coseno

$\int \cos cx\;dx={\frac {1}{c}}\mathrm {sen} \,cx$

$\int \cos ^{n}cx\;dx={\frac {\cos ^{n-1}cx\mathrm {sen} \,cx}{nc}}+{\frac {n-1}{n}}\int \cos ^{n-2}cx\;dx\qquad {\mbox{(per }}n>0{\mbox{)}}$

$\int x\cos cx\;dx={\frac {\cos cx}{c^{2}}}+{\frac {x\mathrm {sen} \,cx}{c}}$

$\int x^{n}\cos cx\;dx={\frac {x^{n}\mathrm {sen} \,cx}{c}}-{\frac {n}{c}}\int x^{n-1}\mathrm {sen} \,cx\;dx$

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

$\int {\frac {\cos cx}{x^{n}}}dx=-{\frac {\cos cx}{(n-1)x^{n-1}}}-{\frac {c}{n-1}}\int {\frac {\mathrm {sen} \,cx}{x^{n-1}}}dx\qquad {\mbox{(per }}n\neq 1{\mbox{)}}$

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

$\int {\frac {dx}{\cos ^{n}cx}}={\frac {\mathrm {sen} \,cx}{c(n-1)cos^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\cos ^{n-2}cx}}\qquad {\mbox{(per }}n>1{\mbox{)}}$

$\int {\frac {dx}{1+\cos cx}}={\frac {1}{c}}\tan {\frac {cx}{2}}$

$\int {\frac {dx}{1-\cos cx}}=-{\frac {1}{c}}\cot {\frac {cx}{2}}$

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

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

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

$\int {\frac {\cos cx\;dx}{1-\cos cx}}=-x-{\frac {1}{c}}\cot {\frac {cx}{2}}$

$\int \cos c_{1}x\cos c_{2}x\;dx={\frac {\mathrm {sen} (c_{1}-c_{2})x}{2(c_{1}-c_{2})}}+{\frac {\mathrm {sen} (c_{1}+c_{2})x}{2(c_{1}+c_{2})}}\qquad {\mbox{(per }}|c_{1}|\neq |c_{2}|{\mbox{)}}$

Integrali di funzioni trigonometriche contenenti solo tangente

$\int \tan cx\;dx=-{\frac {1}{c}}\ln |\cos cx|$

$\int \tan ^{n}cx\;dx={\frac {1}{c(n-1)}}\tan ^{n-1}cx-\int \tan ^{n-2}cx\;dx\qquad {\mbox{(per }}n\neq 1{\mbox{)}}$

$\int {\frac {dx}{\tan cx+1}}={\frac {x}{2}}+{\frac {1}{2c}}\ln |\mathrm {sen} \,cx+\cos cx|$

$\int {\frac {dx}{\tan cx-1}}=-{\frac {x}{2}}+{\frac {1}{2c}}\ln |\mathrm {sen} \,cx-\cos cx|$

$\int {\frac {\tan cx\;dx}{\tan cx+1}}={\frac {x}{2}}-{\frac {1}{2c}}\ln |\mathrm {sen} \,cx+\cos cx|$

$\int {\frac {\tan cx\;dx}{\tan cx-1}}={\frac {x}{2}}+{\frac {1}{2c}}\ln |\mathrm {sen} \,cx-\cos cx|$

Integrali di funzioni trigonometriche contenenti solo secante

$\int \sec {cx}\,dx={\frac {1}{c}}\ln {\left|\sec {cx}+\tan {cx}\right|}$

$\int \sec ^{n}{cx}\,dx={\frac {\sec ^{n-1}{cx}\mathrm {sen} \,{cx}}{c(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \sec ^{n-2}{cx}\,dx\qquad {\mbox{ per }}n\neq 1,\,c\neq 0$

Integrali di funzioni trigonometriche contenenti solo cosecante

$\int \csc {cx}\,dx=-{\frac {1}{c}}\ln {\left|\csc {cx}+\cot {cx}\right|}$

$\int \csc ^{n}{cx}\,dx=-{\frac {\csc ^{n-1}{cx}\cos {cx}}{c(n-1)}}\,+\,{\frac {n-2}{n-1}}\int \csc ^{n-2}{cx}\,dx\qquad {\mbox{ per }}n\neq 1,\,c\neq 0$

Integrali di funzioni trigonometriche contenenti solo cotangente

$\int \cot cx\;dx={\frac {1}{c}}\ln |\mathrm {sen} \,cx|$

$\int \cot ^{n}cx\;dx=-{\frac {1}{c(n-1)}}\cot ^{n-1}cx-\int \cot ^{n-2}cx\;dx\qquad {\mbox{(per }}n\neq 1{\mbox{)}}$

$\int {\frac {dx}{1+\cot cx}}=\int {\frac {\tan cx\;dx}{\tan cx+1}}$

$\int {\frac {dx}{1-\cot cx}}=\int {\frac {\tan cx\;dx}{\tan cx-1}}$

Integrali di funzioni trigonometriche contenenti seno e coseno

$\int {\frac {dx}{\cos cx\pm \mathrm {sen} \,cx}}={\frac {1}{c{\sqrt {2}}}}\ln \left|\tan \left({\frac {cx}{2}}\pm {\frac {\pi }{8}}\right)\right|$

$\int {\frac {dx}{(\cos cx\pm \mathrm {sen} \,cx)^{2}}}={\frac {1}{2c}}\tan \left(cx\mp {\frac {\pi }{4}}\right)$

$\int {\frac {\cos cx\;dx}{\cos cx+\mathrm {sen} \,cx}}={\frac {x}{2}}+{\frac {1}{2c}}\ln \left|\mathrm {sen} \,\mathrm {sen} \,cx+\cos cx\right|$

$\int {\frac {\cos cx\;dx}{\cos cx-\mathrm {sen} \,cx}}={\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\mathrm {sen} \,cx-\cos cx\right|$

$\int {\frac {\mathrm {sen} \,cx\;dx}{\cos cx+\mathrm {sen} \,cx}}={\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\mathrm {sen} \,cx+\cos cx\right|$

$\int {\frac {\mathrm {sen} \,cx\;dx}{\cos cx-\mathrm {sen} \,cx}}=-{\frac {x}{2}}-{\frac {1}{2c}}\ln \left|\mathrm {sen} \,cx-\cos cx\right|$

$\int {\frac {\cos cx\;dx}{\mathrm {sen} \,cx(1+\cos cx)}}=-{\frac {1}{4c}}\tan ^{2}{\frac {cx}{2}}+{\frac {1}{2c}}\ln \left|\tan {\frac {cx}{2}}\right|$

$\int {\frac {\cos cx\;dx}{\mathrm {sen} \,cx(1-\cos cx)}}=-{\frac {1}{4c}}\cot ^{2}{\frac {cx}{2}}-{\frac {1}{2c}}\ln \left|\tan {\frac {cx}{2}}\right|$

$\int {\frac {\mathrm {sen} \,cx\;dx}{\cos cx(1+\mathrm {sen} \,cx)}}={\frac {1}{4c}}\cot ^{2}\left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)+{\frac {1}{2c}}\ln \left|\tan \left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)\right|$

$\int {\frac {\mathrm {sen} \,cx\;dx}{\cos cx(1-\mathrm {sen} \,cx)}}={\frac {1}{4c}}\tan ^{2}\left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)-{\frac {1}{2c}}\ln \left|\tan \left({\frac {cx}{2}}+{\frac {\pi }{4}}\right)\right|$

$\int \mathrm {sen} \,cx\cos cx\;dx={\frac {-1}{2c}}\cos ^{2}cx$

$\int \mathrm {sen} \,c_{1}x\cos c_{2}x\;dx=-{\frac {\cos(c_{1}+c_{2})x}{2(c_{1}+c_{2})}}-{\frac {\cos(c_{1}-c_{2})x}{2(c_{1}-c_{2})}}\qquad {\mbox{(per }}|c_{1}|\neq |c_{2}|{\mbox{)}}$

$\int \mathrm {sen} \,^{n}cx\cos cx\;dx={\frac {1}{c(n+1)}}\mathrm {sen} \,^{n+1}cx\qquad {\mbox{(per }}n\neq 1{\mbox{)}}$

$\int \mathrm {sen} \,cx\cos ^{n}cx\;dx=-{\frac {1}{c(n+1)}}\cos ^{n+1}cx\qquad {\mbox{(per }}n\neq 1{\mbox{)}}$

$\int \mathrm {sen} \,^{n}cx\cos ^{m}cx\;dx=-{\frac {\mathrm {sen} \,^{n-1}cx\cos ^{m+1}cx}{c(n+m)}}+{\frac {n-1}{n+m}}\int \mathrm {sen} \,^{n-2}cx\cos ^{m}cx\;dx\qquad {\mbox{(per }}m,n>0{\mbox{)}}$

anche: $\int \mathrm {sen} \,^{n}cx\cos ^{m}cx\;dx={\frac {\mathrm {sen} \,^{n+1}cx\cos ^{m-1}cx}{c(n+m)}}+{\frac {m-1}{n+m}}\int \mathrm {sen} \,^{n}cx\cos ^{m-2}cx\;dx\qquad {\mbox{(per }}m,n>0{\mbox{)}}$

$\int {\frac {dx}{\mathrm {sen} \,cx\cos cx}}={\frac {1}{c}}\ln \left|\tan cx\right|$

$\int {\frac {dx}{\mathrm {sen} \,cx\cos ^{n}cx}}={\frac {1}{c(n-1)\cos ^{n-1}cx}}+\int {\frac {dx}{\mathrm {sen} \,cx\cos ^{n-2}cx}}\qquad {\mbox{(per }}n\neq 1{\mbox{)}}$

$\int {\frac {dx}{\mathrm {sen} \,^{n}cx\cos cx}}=-{\frac {1}{c(n-1)\mathrm {sen} \,^{n-1}cx}}+\int {\frac {dx}{\mathrm {sen} \,^{n-2}cx\cos cx}}\qquad {\mbox{(per }}n\neq 1{\mbox{)}}$

$\int {\frac {\mathrm {sen} \,cx\;dx}{\cos ^{n}cx}}={\frac {1}{c(n-1)\cos ^{n-1}cx}}\qquad {\mbox{(per }}n\neq 1{\mbox{)}}$

$\int {\frac {\mathrm {sen} \,^{2}cx\;dx}{\cos cx}}=-{\frac {1}{c}}\mathrm {sen} \,cx+{\frac {1}{c}}\ln \left|\tan \left({\frac {\pi }{4}}+{\frac {cx}{2}}\right)\right|$

$\int {\frac {\mathrm {sen} \,^{2}cx\;dx}{\cos ^{n}cx}}={\frac {\mathrm {sen} \,cx}{c(n-1)\cos ^{n-1}cx}}-{\frac {1}{n-1}}\int {\frac {dx}{\cos ^{n-2}cx}}\qquad {\mbox{(per }}n\neq 1{\mbox{)}}$

$\int {\frac {\mathrm {sen} \,^{n}cx\;dx}{\cos cx}}=-{\frac {\mathrm {sen} \,\mathrm {sen} \,^{n-1}cx}{c(n-1)}}+\int {\frac {\mathrm {sen} \,\mathrm {sen} \,^{n-2}cx\;dx}{\cos cx}}\qquad {\mbox{(per }}n\neq 1{\mbox{)}}$

$\int {\frac {\mathrm {sen} \,^{n}cx\;dx}{\cos ^{m}cx}}={\frac {\mathrm {sen} \,^{n+1}cx}{c(m-1)\cos ^{m-1}cx}}-{\frac {n-m+2}{m-1}}\int {\frac {\mathrm {sen} \,^{n}cx\;dx}{\cos ^{m-2}cx}}\qquad {\mbox{(per }}m\neq 1{\mbox{)}}$

anche: $\int {\frac {\mathrm {sen} \,^{n}cx\;dx}{\cos ^{m}cx}}=-{\frac {\mathrm {sen} \,^{n-1}cx}{c(n-m)\cos ^{m-1}cx}}+{\frac {n-1}{n-m}}\int {\frac {\mathrm {sen} \,^{n-2}cx\;dx}{\cos ^{m}cx}}\qquad {\mbox{(per }}m\neq n{\mbox{)}}$

anche: $\int {\frac {\mathrm {sen} \,^{n}cx\;dx}{\cos ^{m}cx}}={\frac {\mathrm {sen} \,^{n-1}cx}{c(m-1)\cos ^{m-1}cx}}-{\frac {n-1}{m-1}}\int {\frac {\mathrm {sen} \,^{n-1}cx\;dx}{\cos ^{m-2}cx}}\qquad {\mbox{(per }}m\neq 1{\mbox{)}}$

$\int {\frac {\cos cx\;dx}{\mathrm {sen} \,^{n}cx}}=-{\frac {1}{c(n-1)\mathrm {sen} \,\mathrm {sen} \,\mathrm {sen} \,^{n-1}cx}}\qquad {\mbox{(per }}n\neq 1{\mbox{)}}$

$\int {\frac {\cos ^{2}cx\;dx}{\mathrm {sen} \,cx}}={\frac {1}{c}}\left(\cos cx+\ln \left|\tan {\frac {cx}{2}}\right|\right)$

$\int {\frac {\cos ^{2}cx\;dx}{\mathrm {sen} \,^{n}cx}}=-{\frac {1}{n-1}}\left({\frac {\cos cx}{cv^{n-1}cx)}}+\int {\frac {dx}{\mathrm {sen} \,^{n-2}cx}}\right)\qquad {\mbox{(per }}n\neq 1{\mbox{)}}$

$\int {\frac {\cos ^{n}cx\;dx}{v^{m}cx}}=-{\frac {\cos ^{n+1}cx}{c(m-1)\mathrm {sen} \,\mathrm {sen} \,\mathrm {sen} \,^{m-1}cx}}-{\frac {n-m-2}{m-1}}\int {\frac {cos^{n}cx\;dx}{\mathrm {sen} \,^{m-2}cx}}\qquad {\mbox{(per }}m\neq 1{\mbox{)}}$

anche: $\int {\frac {\cos ^{n}cx\;dx}{\mathrm {sen} \,^{m}cx}}={\frac {\cos ^{n-1}cx}{c(n-m)\mathrm {sen} \,^{m-1}cx}}+{\frac {n-1}{n-m}}\int {\frac {cos^{n-2}cx\;dx}{\mathrm {sen} \,^{m}cx}}\qquad {\mbox{(per }}m\neq n{\mbox{)}}$

anche: $\int {\frac {\cos ^{n}cx\;dx}{\mathrm {sen} \,\mathrm {sen} \,\mathrm {sen} \,\mathrm {sen} \,\mathrm {sen} \,v\mathrm {sen} \,\mathrm {sen} \,vvvv^{m}cx}}=-{\frac {\cos ^{n-1}cx}{c(m-1)\mathrm {sen} \,^{m-1}cx}}-{\frac {n-1}{m-1}}\int {\frac {cos^{n-2}cx\;dx}{\mathrm {sen} \,^{m-2}cx}}\qquad {\mbox{(per }}m\neq 1{\mbox{)}}$

Integrali di funzioni trigonometriche contenenti seno e tangente

$\int \mathrm {sen} \,\mathrm {sen} \,\mathrm {sen} \,cx\tan cx\;dx={\frac {1}{c}}(\ln |\sec cx+\tan cx|-\mathrm {sen} \,cx)$

$\int {\frac {\tan ^{n}cx\;dx}{\mathrm {sen} \,^{2}cx}}={\frac {1}{c(n-1)}}\tan ^{n-1}(cx)\qquad {\mbox{(per }}n\neq 1{\mbox{)}}$

Integrali di funzioni trigonometriche contenenti cos e tan

$\int {\frac {\tan ^{n}cx\;dx}{\cos ^{2}cx}}={\frac {1}{c(n+1)}}\tan ^{n+1}cx\qquad {\mbox{(per }}n\neq -1{\mbox{)}}$

Integrali di funzioni trigonometriche contenenti sin e cot

$\int {\frac {\cot ^{n}cx\;dx}{\mathrm {sen} \,^{2}cx}}={\frac {-1}{c(n+1)}}\cot ^{n+1}cx\qquad {\mbox{(per }}n\neq -1{\mbox{)}}$

Integrali di funzioni trigonometriche contenenti cos e cot

$\int {\frac {\cot ^{n}cx\;dx}{\cos ^{2}cx}}={\frac {1}{c(1-n)}}\tan ^{1-n}cx\qquad {\mbox{(per }}n\neq 1{\mbox{)}}$

Integrali di funzioni trigonometriche contenenti tangente e cotangente

$\int {\frac {\tan ^{m}(cx)}{\cot ^{n}(cx)}}\;dx={\frac {1}{c(m+n-1)}}\tan ^{m+n-1}(cx)-\int {\frac {\tan ^{m-2}(cx)}{\cot ^{n}(cx)}}\;dx\qquad {\mbox{(per }}m+n\neq 1{\mbox{)}}$