Differenze tra le versioni di "Tavola degli integrali indefiniti di funzioni trigonometriche"

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== Integrali di funzioni trigonometriche contenenti solo [[Senoil (trigonometria)|sin]]seno ==
{{vedi anche|seno (trigonometria)}}
 
: <math>\int\sinmathrm{sen} \, cx\;dx = -\frac{1}{c}\cos cx</math>
 
: <math>\int\sinmathrm{sen}^n cx\;dx = -\frac{\sinmathrm{sen}^{n-1} cx\cos cx}{nc} + \frac{n-1}{n}\int\sinmathrm{sen}^{n-2} cx\;dx \qquad\mbox{(per }n>0\mbox{)}</math>
 
: <math>\int x\sinmathrm{sen} \, cx\;dx = \frac{\sinmathrm{sen} \, cx}{c^2}-\frac{x\cos cx}{c}</math>
 
: <math>\int x^n\sinmathrm{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{)}</math>
 
: <math>\int\frac{\sinmathrm{sen} \, cx}{x} dx = \sum_{i=0}^\infty (-1)^i\frac{(cx)^{2i+1}}{(2i+1)\cdot (2i+1)!}</math>
 
: <math>\int\frac{\sinmathrm{sen} \, cx}{x^n} dx = -\frac{\sinmathrm{sen} \, cx}{(n-1)x^{n-1}} + \frac{c}{n-1}\int\frac{\cos cx}{x^{n-1}} dx</math>
 
: <math>\int\frac{dx}{\sinmathrm{sen} \, cx} = \frac{1}{c}\ln \left|\tan\frac{cx}{2}\right|</math>
 
: <math>\int\frac{dx}{\sinmathrm{sen}^n cx} = \frac{\cos cx}{c(1-n) \sinmathrm{sen}^{n-1} cx}+\frac{n-2}{n-1}\int\frac{dx}{\sinmathrm{sen}^{n-2}cx} \qquad\mbox{(per }n>1\mbox{)}</math>
 
: <math>\int\frac{dx}{1\pm\sinmathrm{sen} \, cx} = \frac{1}{c}\tan\left(\frac{cx}{2}\mp\frac{\pi}{4}\right)</math>
 
: <math>\int\frac{x\;dx}{1+\sinmathrm{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|</math>
 
: <math>\int\frac{x\;dx}{1-\sinmathrm{sen} \, cx} = \frac{x}{c}\cot\left(\frac{\pi}{4} - \frac{cx}{2}\right)+\frac{2}{c^2}\ln\left|\sinmathrm{sen} \, \left(\frac{\pi}{4}-\frac{cx}{2}\right)\right|</math>
 
: <math>\int\frac{\sinmathrm{sen} \, cx\;dx}{1\pm \sinmathrm{sen} \, cx} = \pm x+\frac{1}{c}\tan\left(\frac{\pi}{4}\mp\frac{cx}{2}\right)</math>
 
: <math>\int\sinmathrm{sen} \, c_1x\sinmathrm{sen} \, c_2x\;dx = \frac{\sinmathrm{sen}(c_1-c_2)x}{2(c_1-c_2)}-\frac{\sinmathrm{sen} (c_1+c_2)x}{2(c_1+c_2)} \qquad\mbox{(per }|c_1|\neq|c_2|\mbox{)}</math>
 
== Integrali di funzioni trigonometriche contenenti solo [[Coseno|cos]]il coseno ==
{{vedi anche|coseno}}
 
: <math>\int\cos cx\;dx = \frac{1}{c}\sinmathrm{sen} \, cx</math>
 
: <math>\int x\cos^n\cos cx\;dx = \frac{x\cos^{n-1} cx\sinmathrm{sen} \, cx}{cnc} -+ \frac{n-1}{cn}\int x\cos^{n-12}\sin cx\;dx \qquad\mbox{(per }n>0\mbox{)}</math>
 
: <math>\int x\cos^n cx\;dx = \frac{\cos^{n-1} cx\sin cx}{ncc^2} + \frac{n-1}{n}x\int\cos^mathrm{n-2sen} cx\;dx \qquad\mbox{(per, cx}n>0\mbox{)c}</math>
 
: <math>\int x^n\cos cx\;dx = \frac{x^n\cosmathrm{sen} \, cx}{c^2} +- \frac{xn}{c}\sinint cxx^{n-1}\mathrm{csen} \, cx\;dx</math>
 
: <math>\int x^n\cos cx\;dx = \frac{x^n\sin cx}{c} - \frac{n}{c}\int x^{n-1}\sin cx\;dx</math>
 
: <math>\int\frac{\cos cx}{x} dx = \ln|cx|+\sum_{i=1}^\infty (-1)^i\frac{(cx)^{2i}}{2i\cdot(2i)!}</math>
 
: <math>\int\frac{\cos cx}{x^n} dx = -\frac{\cos cx}{(n-1)x^{n-1}}-\frac{c}{n-1}\int\frac{\sinmathrm{sen} \, cx}{x^{n-1}} dx \qquad\mbox{(per }n\neq 1\mbox{)}</math>
 
: <math>\int\frac{dx}{\cos cx} = \frac{1}{c}\ln\left|\tan\left(\frac{cx}{2}+\frac{\pi}{4}\right)\right|</math>
 
: <math>\int\frac{dx}{\cos^n cx} = \frac{\sinmathrm{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{)}</math>
 
: <math>\int\frac{dx}{1+\cos cx} = \frac{1}{c}\tan\frac{cx}{2}</math>
: <math>\int\frac{x\;dx}{1+\cos cx} = \frac{x}{c}\tan({cx}/{2}) + \frac{2}{c^2}\ln\left|\cos\frac{cx}{2}\right|</math>
 
: <math>\int\frac{x\;dx}{1-\cos cx} = -\frac{x}{x}\cot({cx}/{2})+\frac{2}{c^2}\ln\left|\sinmathrm{sen}\frac{cx}{2}\right|</math>
 
: <math>\int\frac{\cos cx\;dx}{1+\cos cx} = x - \frac{1}{c}\tan\frac{cx}{2}</math>
: <math>\int\frac{\cos cx\;dx}{1-\cos cx} = -x-\frac{1}{c}\cot\frac{cx}{2}</math>
 
: <math>\int\cos c_1x\cos c_2x\;dx = \frac{\sinmathrm{sen}(c_1-c_2)x}{2(c_1-c_2)}+\frac{\sinmathrm{sen}(c_1+c_2)x}{2(c_1+c_2)} \qquad\mbox{(per }|c_1|\neq|c_2|\mbox{)}</math>
 
== Integrali di funzioni trigonometriche contenenti solo [[Tangente (trigonometria)|tan]]tangente ==
{{vedi anche|tangente (trigonometria)}}
 
: <math>\int\tan cx\;dx = -\frac{1}{c}\ln|\cos cx|</math>
: <math>\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{)}</math>
 
: <math>\int\frac{dx}{\tan cx + 1} = \frac{x}{2} + \frac{1}{2c}\ln|\sinmathrm{sen} \, cx + \cos cx|</math>
 
: <math>\int\frac{dx}{\tan cx - 1} = -\frac{x}{2} + \frac{1}{2c}\ln|\sinmathrm{sen} \, cx - \cos cx|</math>
 
: <math>\int\frac{\tan cx\;dx}{\tan cx + 1} = \frac{x}{2} - \frac{1}{2c}\ln|\sinmathrm{sen} \, cx + \cos cx|</math>
 
: <math>\int\frac{\tan cx\;dx}{\tan cx - 1} = \frac{x}{2} + \frac{1}{2c}\ln|\sinmathrm{sen} \, cx - \cos cx|</math>
 
== Integrali di funzioni trigonometriche contenenti solo [[Secante (trigonometria)|sec]]secante ==
{{vedi anche|secante (trigonometria)}}
 
:<math>\int \sec{cx} \, dx = \frac{1}{c}\ln{\left| \sec{cx} + \tan{cx}\right|}</math>
 
:<math>\int \sec^n{cx} \, dx = \frac{\sec^{n-1}{cx} \sinmathrm{sen} \, {cx}}{c(n-1)} \,+\, \frac{n-2}{n-1}\int \sec^{n-2}{cx} \, dx \qquad \mbox{ per }n \ne 1,\,c \ne 0</math>
 
== Integrali di funzioni trigonometriche contenenti solo [[Cosecante|csc]]cosecante ==
{{vedi anche|cosecante (trigonometria)}}
 
:<math>\int \csc{cx} \, dx = -\frac{1}{c}\ln{\left| \csc{cx} + \cot{cx}\right|}</math>
:<math>\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 \ne 1,\,c \ne 0</math>
 
== Integrali di funzioni trigonometriche contenenti solo [[Cotangente|cot]]cotangente ==
{{vedi anche|cotangente (trigonometria)}}
 
: <math>\int\cot cx\;dx = \frac{1}{c}\ln|\sinmathrm{sen} \, cx|</math>
 
: <math>\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{)}</math>
: <math>\int\frac{dx}{1 - \cot cx} = \int\frac{\tan cx\;dx}{\tan cx-1}</math>
 
== Integrali di funzioni trigonometriche contenenti [[Seno (trigonometria)|sin]]seno e [[Coseno|cos]] coseno==
 
 
: <math>\int\frac{dx}{\cos cx\pm\sinmathrm{sen} \, cx} = \frac{1}{c\sqrt{2}}\ln\left|\tan\left(\frac{cx}{2}\pm\frac{\pi}{8}\right)\right|</math>
 
: <math>\int\frac{dx}{(\cos cx\pm\sinmathrm{sen} \, cx)^2} = \frac{1}{2c}\tan\left(cx\mp\frac{\pi}{4}\right)</math>
 
: <math>\int\frac{\cos cx\;dx}{\cos cx + \sinmathrm{sen} \, cx} = \frac{x}{2} + \frac{1}{2c}\ln\left|\sinmathrm{sen} \,\mathrm{sen} \, cx + \cos cx\right|</math>
 
: <math>\int\frac{\cos cx\;dx}{\cos cx - \sinmathrm{sen} \, cx} = \frac{x}{2} - \frac{1}{2c}\ln\left|\sinmathrm{sen} \, cx - \cos cx\right|</math>
 
: <math>\int\frac{\sinmathrm{sen} \, cx\;dx}{\cos cx + \sinmathrm{sen} \, cx} = \frac{x}{2} - \frac{1}{2c}\ln\left|\sinmathrm{sen} \, cx + \cos cx\right|</math>
 
: <math>\int\frac{\sinmathrm{sen} \, cx\;dx}{\cos cx - \sinmathrm{sen} \, cx} = -\frac{x}{2} - \frac{1}{2c}\ln\left|\sinmathrm{sen} \, cx - \cos cx\right|</math>
 
: <math>\int\frac{\cos cx\;dx}{\sinmathrm{sen} \, cx(1+\cos cx)} = -\frac{1}{4c}\tan^2\frac{cx}{2}+\frac{1}{2c}\ln\left|\tan\frac{cx}{2}\right|</math>
 
: <math>\int\frac{\cos cx\;dx}{\sinmathrm{sen} \, cx(1-\cos cx)} = -\frac{1}{4c}\cot^2\frac{cx}{2}-\frac{1}{2c}\ln\left|\tan\frac{cx}{2}\right|</math>
 
: <math>\int\frac{\sinmathrm{sen} \, cx\;dx}{\cos cx(1+\sinmathrm{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|</math>
 
: <math>\int\frac{\sinmathrm{sen} \, cx\;dx}{\cos cx(1-\sinmathrm{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|</math>
 
: <math>\int\sinmathrm{sen} \, cx\cos cx\;dx = \frac{-1}{2c}\cos^2 cx</math>
 
: <math>\int\sinmathrm{sen} \, c_1x\cos c_2x\;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{)}</math>
 
: <math>\int\sinmathrm{sen} \,^n cx\cos cx\;dx = \frac{1}{c(n+1)}\sinmathrm{sen} \,^{n+1} cx \qquad\mbox{(per }n\neq 1\mbox{)}</math>
 
: <math>\int\sinmathrm{sen} \, cx\cos^n cx\;dx = -\frac{1}{c(n+1)}\cos^{n+1} cx \qquad\mbox{(per }n\neq 1\mbox{)}</math>
 
: <math>\int\sinmathrm{sen} \,^n cx\cos^m cx\;dx = -\frac{\sinmathrm{sen} \,^{n-1} cx\cos^{m+1} cx}{c(n+m)}+\frac{n-1}{n+m}\int\sinmathrm{sen} \,^{n-2} cx\cos^m cx\;dx \qquad\mbox{(per }m,n>0\mbox{)}</math>
 
: anche: <math>\int\sinmathrm{sen} \,^n cx\cos^m cx\;dx = \frac{\sinmathrm{sen} \,^{n+1} cx\cos^{m-1} cx}{c(n+m)} + \frac{m-1}{n+m}\int\sinmathrm{sen} \,^n cx\cos^{m-2} cx\;dx \qquad\mbox{(per }m,n>0\mbox{)}</math>
 
: <math>\int\frac{dx}{\sinmathrm{sen} \, cx\cos cx} = \frac{1}{c}\ln\left|\tan cx\right|</math>
 
: <math>\int\frac{dx}{\sinmathrm{sen} \, cx\cos^n cx} = \frac{1}{c(n-1)\cos^{n-1} cx}+\int\frac{dx}{\sinmathrm{sen} \, cx\cos^{n-2} cx} \qquad\mbox{(per }n\neq 1\mbox{)}</math>
 
: <math>\int\frac{dx}{\sinmathrm{sen} \,^n cx\cos cx} = -\frac{1}{c(n-1)\sinmathrm{sen} \,^{n-1} cx}+\int\frac{dx}{\sinmathrm{sen} \,^{n-2} cx\cos cx} \qquad\mbox{(per }n\neq 1\mbox{)}</math>
 
: <math>\int\frac{\sinmathrm{sen} \, cx\;dx}{\cos^n cx} = \frac{1}{c(n-1)\cos^{n-1} cx} \qquad\mbox{(per }n\neq 1\mbox{)}</math>
 
: <math>\int\frac{\sinmathrm{sen} \,^2 cx\;dx}{\cos cx} = -\frac{1}{c}\sinmathrm{sen} \, cx+\frac{1}{c}\ln\left|\tan\left(\frac{\pi}{4}+\frac{cx}{2}\right)\right|</math>
 
: <math>\int\frac{\sinmathrm{sen} \,^2 cx\;dx}{\cos^n cx} = \frac{\sinmathrm{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{)}</math>
 
: <math>\int\frac{\sinmathrm{sen} \,^n cx\;dx}{\cos cx} = -\frac{\sinmathrm{sen} \,\mathrm{sen} \,^{n-1} cx}{c(n-1)} + \int\frac{\sinmathrm{sen} \,\mathrm{sen} \,^{n-2} cx\;dx}{\cos cx} \qquad\mbox{(per }n\neq 1\mbox{)}</math>
 
: <math>\int\frac{\sinmathrm{sen} \,^n cx\;dx}{\cos^m cx} = \frac{\sinmathrm{sen} \,^{n+1} cx}{c(m-1)\cos^{m-1} cx}-\frac{n-m+2}{m-1}\int\frac{\sinmathrm{sen} \,^n cx\;dx}{\cos^{m-2} cx} \qquad\mbox{(per }m\neq 1\mbox{)}</math>
 
: anche: <math>\int\frac{\sinmathrm{sen} \,^n cx\;dx}{\cos^m cx} = -\frac{\sinmathrm{sen} \,^{n-1} cx}{c(n-m)\cos^{m-1} cx}+\frac{n-1}{n-m}\int\frac{\sinmathrm{sen} \,^{n-2} cx\;dx}{\cos^m cx} \qquad\mbox{(per }m\neq n\mbox{)}</math>
 
: anche: <math>\int\frac{\sinmathrm{sen} \,^n cx\;dx}{\cos^m cx} = \frac{\sinmathrm{sen} \,^{n-1} cx}{c(m-1)\cos^{m-1} cx}-\frac{n-1}{m-1}\int\frac{\sinmathrm{sen} \,^{n-1} cx\;dx}{\cos^{m-2} cx} \qquad\mbox{(per }m\neq 1\mbox{)}</math>
 
: <math>\int\frac{\cos cx\;dx}{\sinmathrm{sen} \,^n cx} = -\frac{1}{c(n-1)\sinmathrm{sen} \,\mathrm{sen} \,\mathrm{sen} \,^{n-1} cx} \qquad\mbox{(per }n\neq 1\mbox{)}</math>
 
: <math>\int\frac{\cos^2 cx\;dx}{\sinmathrm{sen} \, cx} = \frac{1}{c}\left(\cos cx+\ln\left|\tan\frac{cx}{2}\right|\right)</math>
 
 
: <math>\int\frac{\cos^2 cx\;dx}{\sinmathrm{sen} \,^n cx} = -\frac{1}{n-1}\left(\frac{\cos cx}{c\sincv^{n-1} cx)}+\int\frac{dx}{\sinmathrm{sen} \,^{n-2} cx}\right) \qquad\mbox{(per }n\neq 1\mbox{)}</math>
 
 
: <math>\int\frac{\cos^n cx\;dx}{\sinv^m cx} = -\frac{\cos^{n+1} cx}{c(m-1)\sinmathrm{sen} \,\mathrm{sen} \,\mathrm{sen} \,^{m-1} cx} - \frac{n-m-2}{m-1}\int\frac{cos^n cx\;dx}{\sinmathrm{sen} \,^{m-2} cx} \qquad\mbox{(per }m\neq 1\mbox{)}</math>
 
 
: anche: <math>\int\frac{\cos^n cx\;dx}{\sinmathrm{sen} \,^m cx} = \frac{\cos^{n-1} cx}{c(n-m)\sinmathrm{sen} \,^{m-1} cx} + \frac{n-1}{n-m}\int\frac{cos^{n-2} cx\;dx}{\sinmathrm{sen} \,^m cx} \qquad\mbox{(per }m\neq n\mbox{)}</math>
 
 
: anche: <math>\int\frac{\cos^n cx\;dx}{\sinmathrm{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)\sinmathrm{sen} \,^{m-1} cx} - \frac{n-1}{m-1}\int\frac{cos^{n-2} cx\;dx}{\sinmathrm{sen} \,^{m-2} cx} \qquad\mbox{(per }m\neq 1\mbox{)}</math>
 
== Integrali di funzioni trigonometriche contenenti [[Seno (trigonometria)|sin]]seno e [[Tangente (trigonometria)|tan]]tangente ==
 
: <math>\int \sinmathrm{sen} \,\mathrm{sen} \,\mathrm{sen} \, cx \tan cx\;dx = \frac{1}{c}(\ln|\sec cx + \tan cx| - \sinmathrm{sen} \, cx)</math>
 
: <math>\int\frac{\tan^n cx\;dx}{\sinmathrm{sen} \,^2 cx} = \frac{1}{c(n-1)}\tan^{n-1} (cx) \qquad\mbox{(per }n\neq 1\mbox{)}</math>
 
== Integrali di funzioni trigonometriche contenenti [[Coseno|cos]] e [[Tangente (trigonometria)|tan]] ==
== Integrali di funzioni trigonometriche contenenti [[Seno (trigonometria)|sin]] e [[Cotangente|cot]] ==
 
: <math>\int\frac{\cot^n cx\;dx}{\sinmathrm{sen} \,^2 cx} = \frac{-1}{c(n+1)}\cot^{n+1} cx \qquad\mbox{(per }n\neq -1\mbox{)}</math>
 
== Integrali di funzioni trigonometriche contenenti [[Coseno|cos]] e [[Tangente (trigonometria)|cot]] ==
: <math>\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{)}</math>
 
== Integrali di funzioni trigonometriche contenenti [[Tangente (trigonometria)|tan]]tangente e [[Cotangente|cot]]cotangente ==
 
: <math>\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{)}</math>