Zerrenda:Funtzio trigonometrikoen integralak

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

\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_{1}x\sin b_{2}x\;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

\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_{1}x\cos a_{2}x\;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

\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

\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\neq 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

\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\neq 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

\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

\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_{1}x\cos a_{2}x\;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

\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

\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

\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

\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

\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

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