Funtzio trigonometrikoen integralen zerrenda

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Ondorengoa funtzio trigonometrikoen integralen zerrenda bat da (jatorrizkoak edo antideribatuak). Funtzio esponentzialak eta trigonometrikoak biak barnean hartzen dituzten jatorrizkoak aztertzeko, ikusi funtzio esponentzialen integralen zerrenda. Integralen zerrenda osatuago nahi baduzu, ikusi integralen zerrenda. Ikusi ere integral trigonometrikoa.

Formula guztietan a konstantea ezin da zero izan, eta K integrazio-konstantea da.

Sinua besterik ez duten integralak[aldatu | aldatu iturburu kodea]

\int\sin ax\;dx = -\frac{1}{a}\cos ax+K\,\!
\int\sin^2 {ax}\;dx = \frac{x}{2} - \frac{1}{4a} \sin 2ax +C= \frac{x}{2} - \frac{1}{2a} \sin ax\cos ax +K\!
\int x\sin^2 {ax}\;dx = \frac{x^2}{4} - \frac{x}{4a} \sin 2ax - \frac{1}{8a^2} \cos 2ax +K\!
\int x^2\sin^2 {ax}\;dx = \frac{x^3}{6} - \left( \frac {x^2}{4a} - \frac{1}{8a^3} \right) \sin 2ax - \frac{x}{4a^2} \cos 2ax +K\!
\int\sin b_1x\sin b_2x\;dx = \frac{\sin((b_1-b_2)x)}{2(b_1-b_2)}-\frac{\sin((b_1+b_2)x)}{2(b_1+b_2)}+K \qquad\mbox{( }|b_1|\neq|b_2|\mbox{)}\,\!
\int\sin^n {ax}\;dx = -\frac{\sin^{n-1} ax\cos ax}{na} + \frac{n-1}{n}\int\sin^{n-2} ax\;dx \qquad\mbox{( }n>0\mbox{)}\,\!
\int\frac{dx}{\sin ax} = \frac{1}{a}\ln \left|\tan\frac{ax}{2}\right|+K
\int\frac{dx}{\sin^n ax} = \frac{\cos ax}{a(1-n) \sin^{n-1} ax}+\frac{n-2}{n-1}\int\frac{dx}{\sin^{n-2}ax} \qquad\mbox{( }n>1\mbox{)}\,\!
\int x\sin ax\;dx = \frac{\sin ax}{a^2}-\frac{x\cos ax}{a}+K\,\!
\int x^n\sin ax\;dx = -\frac{x^n}{a}\cos ax+\frac{n}{a}\int x^{n-1}\cos ax\;dx = \sum_{k=0}^{2k\leq n} (-1)^{k+1} \frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!} \cos ax +\sum_{k=0}^{2k+1\leq n}(-1)^k \frac{x^{n-1-2k}}{a^{2+2k}}\frac{n!}{(n-2k-1)!} \sin ax  \qquad\mbox{( }n>0\mbox{)}\,\!
\int_{\frac{-a}{2}}^{\frac{a}{2}} x^2\sin^2 {\frac{n\pi x}{a}}\;dx = \frac{a^3(n^2\pi^2-6)}{24n^2\pi^2}   \qquad\mbox{( }n=2,4,6...\mbox{)}\,\!
\int\frac{\sin ax}{x} dx = \sum_{n=0}^\infty (-1)^n\frac{(ax)^{2n+1}}{(2n+1)\cdot (2n+1)!} +K\,\!
\int\frac{\sin ax}{x^n} dx = -\frac{\sin ax}{(n-1)x^{n-1}} + \frac{a}{n-1}\int\frac{\cos ax}{x^{n-1}} dx\,\!
\int\frac{dx}{1\pm\sin ax} = \frac{1}{a}\tan\left(\frac{ax}{2}\mp\frac{\pi}{4}\right)+K
\int\frac{x\;dx}{1+\sin ax} = \frac{x}{a}\tan\left(\frac{ax}{2} - \frac{\pi}{4}\right)+\frac{2}{a^2}\ln\left|\cos\left(\frac{ax}{2}-\frac{\pi}{4}\right)\right|+K
\int\frac{x\;dx}{1-\sin ax} = \frac{x}{a}\cot\left(\frac{\pi}{4} - \frac{ax}{2}\right)+\frac{2}{a^2}\ln\left|\sin\left(\frac{\pi}{4}-\frac{ax}{2}\right)\right|+K
\int\frac{\sin ax\;dx}{1\pm\sin ax} = \pm x+\frac{1}{a}\tan\left(\frac{\pi}{4}\mp\frac{ax}{2}\right)+K

Kosinua besterik ez duten integralak[aldatu | aldatu iturburu kodea]

\int\cos ax\;dx = \frac{1}{a}\sin ax+K\,\!
\int\cos^2 {ax}\;dx = \frac{x}{2} + \frac{1}{4a} \sin 2ax +K = \frac{x}{2} + \frac{1}{2a} \sin ax\cos ax +K\!
\int\cos^n ax\;dx = \frac{\cos^{n-1} ax\sin ax}{na} + \frac{n-1}{n}\int\cos^{n-2} ax\;dx \qquad\mbox{( }n>0\mbox{)}\,\!
\int x\cos ax\;dx = \frac{\cos ax}{a^2} + \frac{x\sin ax}{a}+K\,\!
\int x^2\cos^2 {ax}\;dx = \frac{x^3}{6} + \left( \frac {x^2}{4a} - \frac{1}{8a^3} \right) \sin 2ax + \frac{x}{4a^2} \cos 2ax+K\!
\int x^n\cos ax\;dx = \frac{x^n\sin ax}{a} - \frac{n}{a}\int x^{n-1}\sin ax\;dx\,= \sum_{k=0}^{2k+1\leq n} (-1)^{k} \frac{x^{n-2k-1}}{a^{2+2k}}\frac{n!}{(n-2k-1)!} \cos ax +\sum_{k=0}^{2k\leq n}(-1)^{k} \frac{x^{n-2k}}{a^{1+2k}}\frac{n!}{(n-2k)!} \sin ax  \!
\int\frac{\cos ax}{x} dx = \ln|ax|+\sum_{k=1}^\infty (-1)^k\frac{(ax)^{2k}}{2k\cdot(2k)!}+K\,\!
\int\frac{\cos ax}{x^n} dx = -\frac{\cos ax}{(n-1)x^{n-1}}-\frac{a}{n-1}\int\frac{\sin ax}{x^{n-1}} dx \qquad\mbox{( }n\neq 1\mbox{)}\,\!
\int\frac{dx}{\cos ax} = \frac{1}{a}\ln\left|\tan\left(\frac{ax}{2}+\frac{\pi}{4}\right)\right|+K
\int\frac{dx}{\cos^n ax} = \frac{\sin ax}{a(n-1) \cos^{n-1} ax} + \frac{n-2}{n-1}\int\frac{dx}{\cos^{n-2} ax} \qquad\mbox{( }n>1\mbox{)}\,\!
\int\frac{dx}{1+\cos ax} = \frac{1}{a}\tan\frac{ax}{2}+K\,\!
\int\frac{dx}{1-\cos ax} = -\frac{1}{a}\cot\frac{ax}{2}+K\,\!
\int\frac{x\;dx}{1+\cos ax} = \frac{x}{a}\tan\frac{ax}{2} + \frac{2}{a^2}\ln\left|\cos\frac{ax}{2}\right|+K
\int\frac{x\;dx}{1-\cos ax} = -\frac{x}{a}\cot\frac{ax}{2}+\frac{2}{a^2}\ln\left|\sin\frac{ax}{2}\right|+K
\int\frac{\cos ax\;dx}{1+\cos ax} = x - \frac{1}{a}\tan\frac{ax}{2}+K\,\!
\int\frac{\cos ax\;dx}{1-\cos ax} = -x-\frac{1}{a}\cot\frac{ax}{2}+K\,\!
\int\cos a_1x\cos a_2x\;dx = \frac{\sin(a_1-a_2)x}{2(a_1-a_2)}+\frac{\sin(a_1+a_2)x}{2(a_1+a_2)}+K \qquad\mbox{( }|a_1|\neq|a_2|\mbox{)}\,\!

tangentea besterik ez duten integralak[aldatu | aldatu iturburu kodea]

\int\tan ax\;dx = -\frac{1}{a}\ln|\cos ax|+C = \frac{1}{a}\ln|\sec ax|+K\,\!
\int\tan^n ax\;dx = \frac{1}{a(n-1)}\tan^{n-1} ax-\int\tan^{n-2} ax\;dx \qquad\mbox{( }n\neq 1\mbox{)}\,\!
\int\frac{dx}{q \tan ax + p} = \frac{1}{p^2 + q^2}(px + \frac{q}{a}\ln|q\sin ax + p\cos ax|)+K \qquad\mbox{( }p^2 + q^2\neq 0\mbox{)}\,\!


\int\frac{dx}{\tan ax} = \frac{1}{a}\ln|\sin ax|+K\,\!
\int\frac{dx}{\tan ax + 1} = \frac{x}{2} + \frac{1}{2a}\ln|\sin ax + \cos ax|+K\,\!
\int\frac{dx}{\tan ax - 1} = -\frac{x}{2} + \frac{1}{2a}\ln|\sin ax - \cos ax|+K\,\!
\int\frac{\tan ax\;dx}{\tan ax + 1} = \frac{x}{2} - \frac{1}{2a}\ln|\sin ax + \cos ax|+K\,\!
\int\frac{\tan ax\;dx}{\tan ax - 1} = \frac{x}{2} + \frac{1}{2a}\ln|\sin ax - \cos ax|+K\,\!

Sekantea besterik ez duten integralak[aldatu | aldatu iturburu kodea]

\int \sec{ax} \, dx = \frac{1}{a}\ln{\left| \sec{ax} + \tan{ax}\right|}+K
\int \sec^2{x} \, dx = \tan{x}+K
\int \sec^n{ax} \, dx = \frac{\sec^{n-1}{ax} \sin {ax}}{a(n-1)} \,+\, \frac{n-2}{n-1}\int \sec^{n-2}{ax} \, dx \qquad \mbox{ ( }n \ne 1\mbox{)}\,\!
\int \sec^n{x} \, dx = \frac{\sec^{n-2}{x}\tan{x}}{n-1} \,+\, \frac{n-2}{n-1}\int \sec^{n-2}{x}\,dx[1]
\int \frac{dx}{\sec{x} + 1} = x - \tan{\frac{x}{2}}+K
\int \frac{dx}{\sec{x} - 1} = - x - \cot{\frac{x}{2}}+K

Kosekantea besterik ez duten integralak[aldatu | aldatu iturburu kodea]

\int \csc{ax} \, dx = -\frac{1}{a}\ln{\left| \csc{ax} + \cot{ax}\right|}+K
\int \csc^2{x} \, dx = -\cot{x}+K
\int \csc^n{ax} \, dx = -\frac{\csc^{n-1}{ax} \cos{ax}}{a(n-1)} \,+\, \frac{n-2}{n-1}\int \csc^{n-2}{ax} \, dx \qquad \mbox{ ( }n \ne 1\mbox{)}\,\!
\int \frac{dx}{\csc{x} + 1} = x - \frac{2\sin{\frac{x}{2}}}{\cos{\frac{x}{2}}+\sin{\frac{x}{2}}}+K
\int \frac{dx}{\csc{x} - 1} = \frac{2\sin{\frac{x}{2}}}{\cos{\frac{x}{2}}-\sin{\frac{x}{2}}}-x+K

Kotangentea besterik ez duten integralak[aldatu | aldatu iturburu kodea]

\int\cot ax\;dx = \frac{1}{a}\ln|\sin ax|+K\,\!
\int\cot^n ax\;dx = -\frac{1}{a(n-1)}\cot^{n-1} ax - \int\cot^{n-2} ax\;dx \qquad\mbox{( }n\neq 1\mbox{)}\,\!
\int\frac{dx}{1 + \cot ax} = \int\frac{\tan ax\;dx}{\tan ax+1}\,\!
\int\frac{dx}{1 - \cot ax} = \int\frac{\tan ax\;dx}{\tan ax-1}\,\!

Sinua eta kosinua besterik ez dituzten integralak[aldatu | aldatu iturburu kodea]

\int\frac{dx}{\cos ax\pm\sin ax} = \frac{1}{a\sqrt{2}}\ln\left|\tan\left(\frac{ax}{2}\pm\frac{\pi}{8}\right)\right|+K
\int\frac{dx}{(\cos ax\pm\sin ax)^2} = \frac{1}{2a}\tan\left(ax\mp\frac{\pi}{4}\right)+K
\int\frac{dx}{(\cos x + \sin x)^n} = \frac{1}{n-1}\left(\frac{\sin x - \cos x}{(\cos x + \sin x)^{n - 1}} - 2(n - 2)\int\frac{dx}{(\cos x + \sin x)^{n-2}} \right)
\int\frac{\cos ax\;dx}{\cos ax + \sin ax} = \frac{x}{2} + \frac{1}{2a}\ln\left|\sin ax + \cos ax\right|+K
\int\frac{\cos ax\;dx}{\cos ax - \sin ax} = \frac{x}{2} - \frac{1}{2a}\ln\left|\sin ax - \cos ax\right|+K
\int\frac{\sin ax\;dx}{\cos ax + \sin ax} = \frac{x}{2} - \frac{1}{2a}\ln\left|\sin ax + \cos ax\right|+K
\int\frac{\sin ax\;dx}{\cos ax - \sin ax} = -\frac{x}{2} - \frac{1}{2a}\ln\left|\sin ax - \cos ax\right|+K
\int\frac{\cos ax\;dx}{\sin ax(1+\cos ax)} = -\frac{1}{4a}\tan^2\frac{ax}{2}+\frac{1}{2a}\ln\left|\tan\frac{ax}{2}\right|+K
\int\frac{\cos ax\;dx}{\sin ax(1-\cos ax)} = -\frac{1}{4a}\cot^2\frac{ax}{2}-\frac{1}{2a}\ln\left|\tan\frac{ax}{2}\right|+K
\int\frac{\sin ax\;dx}{\cos ax(1+\sin ax)} = \frac{1}{4a}\cot^2\left(\frac{ax}{2}+\frac{\pi}{4}\right)+\frac{1}{2a}\ln\left|\tan\left(\frac{ax}{2}+\frac{\pi}{4}\right)\right|+K
\int\frac{\sin ax\;dx}{\cos ax(1-\sin ax)} = \frac{1}{4a}\tan^2\left(\frac{ax}{2}+\frac{\pi}{4}\right)-\frac{1}{2a}\ln\left|\tan\left(\frac{ax}{2}+\frac{\pi}{4}\right)\right|+K
\int\sin ax\cos ax\;dx = -\frac{1}{2a}\cos^2 ax +K\,\!
\int\sin a_1x\cos a_2x\;dx = -\frac{\cos((a_1-a_2)x)}{2(a_1-a_2)} -\frac{\cos((a_1+a_2)x)}{2(a_1+a_2)} +K\qquad\mbox{( }|a_1|\neq|a_2|\mbox{)}\,\!
\int\sin^n ax\cos ax\;dx = \frac{1}{a(n+1)}\sin^{n+1} ax +K\qquad\mbox{( }n\neq -1\mbox{)}\,\!
\int\sin ax\cos^n ax\;dx = -\frac{1}{a(n+1)}\cos^{n+1} ax +K\qquad\mbox{( }n\neq -1\mbox{)}\,\!
\int\sin^n ax\cos^m ax\;dx = -\frac{\sin^{n-1} ax\cos^{m+1} ax}{a(n+m)}+\frac{n-1}{n+m}\int\sin^{n-2} ax\cos^m ax\;dx  \qquad\mbox{( }m,n>0\mbox{)}\,\!
also: \int\sin^n ax\cos^m ax\;dx = \frac{\sin^{n+1} ax\cos^{m-1} ax}{a(n+m)} + \frac{m-1}{n+m}\int\sin^n ax\cos^{m-2} ax\;dx \qquad\mbox{( }m,n>0\mbox{)}\,\!
\int\frac{dx}{\sin ax\cos ax} = \frac{1}{a}\ln\left|\tan ax\right|+K
\int\frac{dx}{\sin ax\cos^n ax} = \frac{1}{a(n-1)\cos^{n-1} ax}+\int\frac{dx}{\sin ax\cos^{n-2} ax} \qquad\mbox{( }n\neq 1\mbox{)}\,\!
\int\frac{dx}{\sin^n ax\cos ax} = -\frac{1}{a(n-1)\sin^{n-1} ax}+\int\frac{dx}{\sin^{n-2} ax\cos ax} \qquad\mbox{( }n\neq 1\mbox{)}\,\!
\int\frac{\sin ax\;dx}{\cos^n ax} = \frac{1}{a(n-1)\cos^{n-1} ax} +C\qquad\mbox{( }n\neq 1\mbox{)}\,\!
\int\frac{\sin^2 ax\;dx}{\cos ax} = -\frac{1}{a}\sin ax+\frac{1}{a}\ln\left|\tan\left(\frac{\pi}{4}+\frac{ax}{2}\right)\right|+K
\int\frac{\sin^2 ax\;dx}{\cos^n ax} = \frac{\sin ax}{a(n-1)\cos^{n-1}ax}-\frac{1}{n-1}\int\frac{dx}{\cos^{n-2}ax} \qquad\mbox{( }n\neq 1\mbox{)}\,\!
\int\frac{\sin^n ax\;dx}{\cos ax} = -\frac{\sin^{n-1} ax}{a(n-1)} + \int\frac{\sin^{n-2} ax\;dx}{\cos ax} \qquad\mbox{( }n\neq 1\mbox{)}\,\!
\int\frac{\sin^n ax\;dx}{\cos^m ax} = \frac{\sin^{n+1} ax}{a(m-1)\cos^{m-1} ax}-\frac{n-m+2}{m-1}\int\frac{\sin^n ax\;dx}{\cos^{m-2} ax} \qquad\mbox{( }m\neq 1\mbox{)}\,\!
also: \int\frac{\sin^n ax\;dx}{\cos^m ax} = -\frac{\sin^{n-1} ax}{a(n-m)\cos^{m-1} ax}+\frac{n-1}{n-m}\int\frac{\sin^{n-2} ax\;dx}{\cos^m ax} \qquad\mbox{( }m\neq n\mbox{)}\,\!
also: \int\frac{\sin^n ax\;dx}{\cos^m ax} = \frac{\sin^{n-1} ax}{a(m-1)\cos^{m-1} ax}-\frac{n-1}{m-1}\int\frac{\sin^{n-2} ax\;dx}{\cos^{m-2} ax} \qquad\mbox{( }m\neq 1\mbox{)}\,\!
\int\frac{\cos ax\;dx}{\sin^n ax} = -\frac{1}{a(n-1)\sin^{n-1} ax} +K\qquad\mbox{( }n\neq 1\mbox{)}\,\!
\int\frac{\cos^2 ax\;dx}{\sin ax} = \frac{1}{a}\left(\cos ax+\ln\left|\tan\frac{ax}{2}\right|\right) +K
\int\frac{\cos^2 ax\;dx}{\sin^n ax} = -\frac{1}{n-1}\left(\frac{\cos ax}{a\sin^{n-1} ax)}+\int\frac{dx}{\sin^{n-2} ax}\right) \qquad\mbox{( }n\neq 1\mbox{)}
\int\frac{\cos^n ax\;dx}{\sin^m ax} = -\frac{\cos^{n+1} ax}{a(m-1)\sin^{m-1} ax} - \frac{n-m-2}{m-1}\int\frac{\cos^n ax\;dx}{\sin^{m-2} ax} \qquad\mbox{( }m\neq 1\mbox{)}\,\!
also: \int\frac{\cos^n ax\;dx}{\sin^m ax} = \frac{\cos^{n-1} ax}{a(n-m)\sin^{m-1} ax} + \frac{n-1}{n-m}\int\frac{\cos^{n-2} ax\;dx}{\sin^m ax} \qquad\mbox{( }m\neq n\mbox{)}\,\!
also: \int\frac{\cos^n ax\;dx}{\sin^m ax} = -\frac{\cos^{n-1} ax}{a(m-1)\sin^{m-1} ax} - \frac{n-1}{m-1}\int\frac{\cos^{n-2} ax\;dx}{\sin^{m-2} ax} \qquad\mbox{( }m\neq 1\mbox{)}\,\!

Sinua eta tangentea besterik ez dituzten integralak[aldatu | aldatu iturburu kodea]

\int \sin ax \tan ax\;dx = \frac{1}{a}(\ln|\sec ax + \tan ax| - \sin ax)+K\,\!
\int\frac{\tan^n ax\;dx}{\sin^2 ax} = \frac{1}{a(n-1)}\tan^{n-1} (ax) +K\qquad\mbox{( }n\neq 1\mbox{)}\,\!

Kosinua eta tangentea besterik ez dituzten integralak[aldatu | aldatu iturburu kodea]

\int\frac{\tan^n ax\;dx}{\cos^2 ax} = \frac{1}{a(n+1)}\tan^{n+1} ax +K\qquad\mbox{( }n\neq -1\mbox{)}\,\!

Sinua eta kotangentea besterik ez dituzten integralak[aldatu | aldatu iturburu kodea]

\int\frac{\cot^n ax\;dx}{\sin^2 ax} = \frac{1}{a(n+1)}\cot^{n+1} ax  +K\qquad\mbox{( }n\neq -1\mbox{)}\,\!

Kosinua eta kotangentea besterik ez dituzten integralak[aldatu | aldatu iturburu kodea]

\int\frac{\cot^n ax\;dx}{\cos^2 ax} = \frac{1}{a(1-n)}\tan^{1-n} ax +K\qquad\mbox{( }n\neq 1\mbox{)}\,\!

Limite simetrikoak dituzten integralak[aldatu | aldatu iturburu kodea]

\int_{{-c}}^{{c}}\sin {x}\;dx = 0 \!
\int_{{-c}}^{{c}}\cos {x}\;dx = 2\int_{{0}}^{{c}}\cos {x}\;dx = 2\int_{{-c}}^{{0}}\cos {x}\;dx = 2\sin {c} \!
\int_{{-c}}^{{c}}\tan {x}\;dx = 0 \!
\int_{-\frac{a}{2}}^{\frac{a}{2}} x^2\cos^2 {\frac{n\pi x}{a}}\;dx = \frac{a^3(n^2\pi^2-6)}{24n^2\pi^2}   \qquad\mbox{( }n=1,3,5...\mbox{)}\,\!

Erreferentziak eta oharrak[aldatu | aldatu iturburu kodea]

  1. Stewart, James. Calculus: Early Transcendentals, 6. Edizioa. Thomson: 2008