Wikipedia, Entziklopedia askea
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.
![{\displaystyle \int \sin ax\;dx=-{\frac {1}{a}}\cos ax+K\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6512fd6b5e4b2cc855bb53a5d51d90d1121529f9)
![{\displaystyle \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\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dcccb1fa28686f2a9d2469a8541ba100d1ee07f)
![{\displaystyle \int x\sin ^{2}{ax}\;dx={\frac {x^{2}}{4}}-{\frac {x}{4a}}\sin 2ax-{\frac {1}{8a^{2}}}\cos 2ax+K\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c04208aac9555cea410f111d2683aac988ec896)
![{\displaystyle \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\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd5a6dab7cee81b436ef11aa96df85d307eb0ead)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c0a7bff2533487b836bc355cd55e658a322602b)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5bde8bd0e01e250d9d31652b30d7827aaa8a425)
![{\displaystyle \int {\frac {dx}{\sin ax}}={\frac {1}{a}}\ln \left|\tan {\frac {ax}{2}}\right|+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22efa07d32ed78e624ae2edf2ea61f0e3eeffb1b)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/381bc5a49149fe4fc3c59cf6907eb756b02f26c0)
![{\displaystyle \int x\sin ax\;dx={\frac {\sin ax}{a^{2}}}-{\frac {x\cos ax}{a}}+K\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f122964d68bba5d01625ecda42aa139182e6ff1)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7846f4f2031fca489b0d4c071986f9762395e1c8)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a73c1c88b847bec210d6af7825627943ca00787f)
![{\displaystyle \int {\frac {\sin ax}{x}}dx=\sum _{n=0}^{\infty }(-1)^{n}{\frac {(ax)^{2n+1}}{(2n+1)\cdot (2n+1)!}}+K\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/beddffd72ddfb192c671c9ee42a89ebac8a15fad)
![{\displaystyle \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\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b03e7a44d70f97f2ffc9285a1cac5ef28d06f483)
![{\displaystyle \int {\frac {dx}{1\pm \sin ax}}={\frac {1}{a}}\tan \left({\frac {ax}{2}}\mp {\frac {\pi }{4}}\right)+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd746f03cf0537075bd3892344c57432856d66a1)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9324ce1887cecb2351adfae2e07d4c8d86561ea7)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21193afc55d8044199049401f4d7b6563014da87)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b4fc4472f75482c10d1db7004516085d18f85fd)
![{\displaystyle \int \cos ax\;dx={\frac {1}{a}}\sin ax+K\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8b02010b7bd8ffd36718867085eb36dce29802b)
![{\displaystyle \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\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/50a6f8d9944cedb8d3be9298622381914c57fc05)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4408d910576537814c44ab8bba718dd9f743ca1)
![{\displaystyle \int x\cos ax\;dx={\frac {\cos ax}{a^{2}}}+{\frac {x\sin ax}{a}}+K\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/804c5ef85feb33865ac43d8ceabba7c8f0283498)
![{\displaystyle \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\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35d35d37080530a2ad59253a7b7f46c11b214378)
![{\displaystyle \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\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7457aa1d1067e0c5b8838fd3824f2d949bf064ad)
![{\displaystyle \int {\frac {\cos ax}{x}}dx=\ln |ax|+\sum _{k=1}^{\infty }(-1)^{k}{\frac {(ax)^{2k}}{2k\cdot (2k)!}}+K\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de9ef0cc47f5d16ffa00fd48ce56e6c47cf189b2)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77bb62fde0fd44d1e84d59f1970da02310cafbcd)
![{\displaystyle \int {\frac {dx}{\cos ax}}={\frac {1}{a}}\ln \left|\tan \left({\frac {ax}{2}}+{\frac {\pi }{4}}\right)\right|+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c085553f7088dde81c1d028d09eb8c9c7cdf744f)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5f1c55eb7d3f1e6e52a364c5097d83cef833d53)
![{\displaystyle \int {\frac {dx}{1+\cos ax}}={\frac {1}{a}}\tan {\frac {ax}{2}}+K\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d18c7485cd14db2877fed866fbc23bbb8979c68d)
![{\displaystyle \int {\frac {dx}{1-\cos ax}}=-{\frac {1}{a}}\cot {\frac {ax}{2}}+K\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52e5ffc2bd4b9beb094963f2a3f5e3a459085430)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9074411304994eeece308bb1c9764c7d15d06067)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b0584b4688ae1ae72443e3ced061565c221d07e)
![{\displaystyle \int {\frac {\cos ax\;dx}{1+\cos ax}}=x-{\frac {1}{a}}\tan {\frac {ax}{2}}+K\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c73a7749538cf3c0f800215ce2095348b2b57e46)
![{\displaystyle \int {\frac {\cos ax\;dx}{1-\cos ax}}=-x-{\frac {1}{a}}\cot {\frac {ax}{2}}+K\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8baaede54f2d62b187ccc630259953d355aae3f3)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc5dd978ed7dcd90b5cf00fa6bf7b9232d0fcc80)
![{\displaystyle \int \tan ax\;dx=-{\frac {1}{a}}\ln |\cos ax|+C={\frac {1}{a}}\ln |\sec ax|+K\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/281ca26571bd4271176a3c109975a97c283078b5)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ecc4fddcd197ce211c39e7429e8d35be5aa6c59b)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/138c653cea1f9b5459a4b6e003400fd8e25b29bd)
![{\displaystyle \int {\frac {dx}{\tan ax}}={\frac {1}{a}}\ln |\sin ax|+K\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea4864d04b6d94260a05ae3f8b5d2c3eb1a49479)
![{\displaystyle \int {\frac {dx}{\tan ax+1}}={\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax+\cos ax|+K\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/876b0c190f0820531e8fed730e16141334de7d4a)
![{\displaystyle \int {\frac {dx}{\tan ax-1}}=-{\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax-\cos ax|+K\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/112ec62db9c955b12639f5a4cf42a7ccfc40fdb2)
![{\displaystyle \int {\frac {\tan ax\;dx}{\tan ax+1}}={\frac {x}{2}}-{\frac {1}{2a}}\ln |\sin ax+\cos ax|+K\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/000f97fc3fe10e486f35380303c71d289a84ac8d)
![{\displaystyle \int {\frac {\tan ax\;dx}{\tan ax-1}}={\frac {x}{2}}+{\frac {1}{2a}}\ln |\sin ax-\cos ax|+K\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77c52f21a87f8479403b1c5b25ffcd40a53e6024)
![{\displaystyle \int \sec {ax}\,dx={\frac {1}{a}}\ln {\left|\sec {ax}+\tan {ax}\right|}+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c43fc72cc8ea640cf6df711454d612882129562e)
![{\displaystyle \int \sec ^{2}{x}\,dx=\tan {x}+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90c8fa877326c25c77d29f13870502ae073ef1df)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32b7cdb4d8d849c468add52e28ef1051c07f4848)
[1]
![{\displaystyle \int {\frac {dx}{\sec {x}+1}}=x-\tan {\frac {x}{2}}+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3d9d10c18b9b400bbca3b0e6561689d47b528d0)
![{\displaystyle \int {\frac {dx}{\sec {x}-1}}=-x-\cot {\frac {x}{2}}+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aef421b6777dfce74011968fef3a6d6d9d3feb17)
![{\displaystyle \int \csc {ax}\,dx=-{\frac {1}{a}}\ln {\left|\csc {ax}+\cot {ax}\right|}+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/533bafc648b2aa9079a93d04e6fd98c1e668f4ff)
![{\displaystyle \int \csc ^{2}{x}\,dx=-\cot {x}+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4d266a6915c38a58aa4e1c307d476e57160a81a)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e9b414347aac261bbbc5cd1243ae79b5c750e8d)
![{\displaystyle \int {\frac {dx}{\csc {x}+1}}=x-{\frac {2\sin {\frac {x}{2}}}{\cos {\frac {x}{2}}+\sin {\frac {x}{2}}}}+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f1ab9f4f8befe50c95cc25671eb2cce7e0a5a5c)
![{\displaystyle \int {\frac {dx}{\csc {x}-1}}={\frac {2\sin {\frac {x}{2}}}{\cos {\frac {x}{2}}-\sin {\frac {x}{2}}}}-x+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67241f41f72de9298aaf44f8ac9cd00f4f008b90)
![{\displaystyle \int \cot ax\;dx={\frac {1}{a}}\ln |\sin ax|+K\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bf8a5adf0346b6a935289a9d57eba0541ba97a5)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3051c855703a1e951d4d2b2a60008fbbf03408)
![{\displaystyle \int {\frac {dx}{1+\cot ax}}=\int {\frac {\tan ax\;dx}{\tan ax+1}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb992449b8d41c9ad694ebc79755b1bc6f05b721)
![{\displaystyle \int {\frac {dx}{1-\cot ax}}=\int {\frac {\tan ax\;dx}{\tan ax-1}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae83987f4461f7cefa25e65f6583476d6f6aa1a7)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/150affe2bd3830e52850c5f4bc9b25321add3c6f)
![{\displaystyle \int {\frac {dx}{(\cos ax\pm \sin ax)^{2}}}={\frac {1}{2a}}\tan \left(ax\mp {\frac {\pi }{4}}\right)+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b9b0ffa290bbbbefa47f335c000629de90546f6)
![{\displaystyle \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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae4a31631ace2155c341f3a42e944454f4d2525b)
![{\displaystyle \int {\frac {\cos ax\;dx}{\cos ax+\sin ax}}={\frac {x}{2}}+{\frac {1}{2a}}\ln \left|\sin ax+\cos ax\right|+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e677fc07688332f8a0eaaa265fa34df40386b3c3)
![{\displaystyle \int {\frac {\cos ax\;dx}{\cos ax-\sin ax}}={\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax-\cos ax\right|+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57b793be758a3b40b81b36ca7d89a7da5603ae47)
![{\displaystyle \int {\frac {\sin ax\;dx}{\cos ax+\sin ax}}={\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax+\cos ax\right|+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e64078444d2d1f52f3f8a91d195b8c85d55508b)
![{\displaystyle \int {\frac {\sin ax\;dx}{\cos ax-\sin ax}}=-{\frac {x}{2}}-{\frac {1}{2a}}\ln \left|\sin ax-\cos ax\right|+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a96c2c9d7da4852e26e70ed9ad3607a0992b0552)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d391fc81b77e30bf7c0c04d3b2eb42403e1f48db)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0bdcd88789e238e37d9fab4e906453ff5f7ba22)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/acd1cca1fe5dd1b9c078052df2d6a171adc382c3)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/422e13d8d39d1647db9ecbcdebdfe5b24a0883df)
![{\displaystyle \int \sin ax\cos ax\;dx=-{\frac {1}{2a}}\cos ^{2}ax+K\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35f7209f05b98b04636374c89754fbaced7e68aa)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ca2905d652e27836b28c9b9f27c73ee39b188e7)
![{\displaystyle \int \sin ^{n}ax\cos ax\;dx={\frac {1}{a(n+1)}}\sin ^{n+1}ax+K\qquad {\mbox{( }}n\neq -1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ceca9d3c62ccc38a2b1a555d41131eed78a22e92)
![{\displaystyle \int \sin ax\cos ^{n}ax\;dx=-{\frac {1}{a(n+1)}}\cos ^{n+1}ax+K\qquad {\mbox{( }}n\neq -1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2baf07f75857d0147c1eba4dd4274473341ada9c)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/631921f2166e4cb0e2e0c13ee7d81cb5518c96c9)
- also:
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ead70425652b6b9178a7c62e4e632a6b54641741)
![{\displaystyle \int {\frac {dx}{\sin ax\cos ax}}={\frac {1}{a}}\ln \left|\tan ax\right|+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec9cbf4479747e34f31dfa43b814572e4af698a5)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9999527d9c91931c9684c457dc11bb9237ba3f78)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f058b4e10ec4a428c3ef94443db6a61f7ba94c5)
![{\displaystyle \int {\frac {\sin ax\;dx}{\cos ^{n}ax}}={\frac {1}{a(n-1)\cos ^{n-1}ax}}+C\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a16349a842734b6d16c75bbad51a485fdd897656)
![{\displaystyle \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}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8825db2967c85445bf23257754e0facd28b1238e)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ae375f02acbc13330f4f7ca1174907bce2e42a6)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cbb8a2d5552d32f2d8d5179a4de2f61fed5e7f1f)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39f02b8a27bbfc2d9bb954463803ad90ed89b11c)
- also:
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8482d91755cb89ec5be7fcd0e7cf3d45a1b7b636)
- also:
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c62c5ff28294ecf73b924cc26cb27b433f71057)
![{\displaystyle \int {\frac {\cos ax\;dx}{\sin ^{n}ax}}=-{\frac {1}{a(n-1)\sin ^{n-1}ax}}+K\qquad {\mbox{( }}n\neq 1{\mbox{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90961f4c9af416407685ae36789dfbc4b2e36155)
![{\displaystyle \int {\frac {\cos ^{2}ax\;dx}{\sin ax}}={\frac {1}{a}}\left(\cos ax+\ln \left|\tan {\frac {ax}{2}}\right|\right)+K}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a94c695811f26fd25d246722d628da97624be56)
![{\displaystyle \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{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/834276438467557365ce2f747b84c21926d01f9f)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cd69f0e0a8f48aeb34375e2a53dad6d1df63f6b)
- also:
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c700a4f61ed4c170266c31cb9e9ccef631ef914)
- also:
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/592ce2f68343d684c8154b0307b77372c3b6750e)
Sinua eta tangentea besterik ez dituzten integralak[aldatu | aldatu iturburu kodea]
![{\displaystyle \int \sin ax\tan ax\;dx={\frac {1}{a}}(\ln |\sec ax+\tan ax|-\sin ax)+K\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5fdf5079dff3b8cb8f748b695b882af7e99350a6)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/715604cbe05dbcf69c7e9efce763de576b1ca765)
Kosinua eta tangentea besterik ez dituzten integralak[aldatu | aldatu iturburu kodea]
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12d6d3128155042d6c9e043b8a662c51305efcd4)
Sinua eta kotangentea besterik ez dituzten integralak[aldatu | aldatu iturburu kodea]
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/772ba521b3a76e8803eaabe11b309ac98a64b8a7)
Kosinua eta kotangentea besterik ez dituzten integralak[aldatu | aldatu iturburu kodea]
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b3a3b14002e65373f6111d443df639c58edbe5f)
![{\displaystyle \int _{-c}^{c}\sin {x}\;dx=0\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33469f374c9c3af903d9607671dcfdf18c1a5077)
![{\displaystyle \int _{-c}^{c}\cos {x}\;dx=2\int _{0}^{c}\cos {x}\;dx=2\int _{-c}^{0}\cos {x}\;dx=2\sin {c}\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43f875b81a6b5c38d2d0a7f54fbe9ae958b47526)
![{\displaystyle \int _{-c}^{c}\tan {x}\;dx=0\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46a562f35f07612ecdd0e562ba5e93b728320032)
![{\displaystyle \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{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02aca4342e4a4ea6013016a157968cd1f4addcc7)
- ↑ Stewart, James. Calculus: Early Transcendentals, 6. Edizioa. Thomson: 2008