من ويكيبيديا، الموسوعة الحرة
هذه قائمة تكاملات الدوال الزائدية.أخذا بالعلم أن
عدد غير منعدم وأن
هي ثابت التكامل.
التكاملات التي تحتوي على دالة الجيب الزائدية[عدل]
![{\displaystyle \int \sinh ax\,dx={\frac {1}{a}}\cosh ax+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f9d9cbcd54f578ac416cd60b4f7f6652ecd3857)
![{\displaystyle \int \sinh ^{2}ax\,dx={\frac {1}{4a}}\sinh 2ax-{\frac {x}{2}}+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/337555e88add80a6b3c2a301058a25aa9f42d99f)
![{\displaystyle \int \sinh ^{n}ax\,dx={\frac {1}{an}}\sinh ^{n-1}ax\cosh ax-{\frac {n-1}{n}}\int \sinh ^{n-2}ax\,dx\qquad {\mbox{(}}n>0{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3909b2e546ab61b7ad8069d5bbe94685b486c6fe)
![{\displaystyle \int \sinh ^{n}ax\,dx={\frac {1}{a(n+1)}}\sinh ^{n+1}ax\cosh ax-{\frac {n+2}{n+1}}\int \sinh ^{n+2}ax\,dx\qquad {\mbox{(}}n<0{\mbox{, }}n\neq -1{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c020737702f0b3b0fbdfa8a72a85931513eb33f0)
![{\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|\tanh {\frac {ax}{2}}\right|+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d14b674a9b740e8c1fe8b3a534365953d3993dd4)
![{\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|{\frac {\cosh ax-1}{\sinh ax}}\right|+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a8d31337ed2598b6933cb5ea53b9a77c44e139c)
![{\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|{\frac {\sinh ax}{\cosh ax+1}}\right|+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9abc694d34204ecd7ccd4af1576834fe16bf8387)
![{\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|{\frac {\cosh ax-1}{\cosh ax+1}}\right|+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de6be9c20a076cde617d6bf30efa385c7023ee1c)
![{\displaystyle \int {\frac {dx}{\sinh ^{n}ax}}=-{\frac {\cosh ax}{a(n-1)\sinh ^{n-1}ax}}-{\frac {n-2}{n-1}}\int {\frac {dx}{\sinh ^{n-2}ax}}\qquad {\mbox{(}}n\neq 1{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e2e7f58a72628da8531e9126a75db8b754e53c0)
![{\displaystyle \int x\sinh ax\,dx={\frac {1}{a}}x\cosh ax-{\frac {1}{a^{2}}}\sinh ax+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ed9225606cd8ddb06324343904e147498176a9d)
![{\displaystyle \int \sinh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8c5604773c7623b9e6a14d17b8cb8ae8cfcf1c8)
![{\displaystyle \int \sinh(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e3f7dfbb4f90692f6804c58f8ecfc2850712af9)
![{\displaystyle \int \sinh ax\sinh bx\,dx={\frac {1}{a^{2}-b^{2}}}(a\sinh bx\cosh ax-b\cosh bx\sinh ax)+C\qquad {\mbox{(}}a^{2}\neq b^{2}{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f675ae2b77ac8122fdfcbc8f019a180c383bae97)
التكاملات التي تحتوي على دالة جيب التمام الزائدية[عدل]
![{\displaystyle \int \cosh ax\,dx={\frac {1}{a}}\sinh ax+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4caebbe3dfd38756272c4f537239370a08d83748)
![{\displaystyle \int \cosh ^{2}ax\,dx={\frac {1}{4a}}\sinh 2ax+{\frac {x}{2}}+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca5f33eab62decd4b3602955da2ec0a23e09e9a2)
![{\displaystyle \int {\frac {dx}{\cosh ax}}={\frac {2}{a}}\arctan e^{ax}+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0dff59f32e29e40ac43a4f357101cdcc9792acf)
![{\displaystyle \int \cosh ^{n}ax\,dx={\frac {1}{an}}\sinh ax\cosh ^{n-1}ax+{\frac {n-1}{n}}\int \cosh ^{n-2}ax\,dx\qquad {\mbox{(}}n>0{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db15390fd20d370cfc11a51bd2e98c38610dc146)
![{\displaystyle \int \cosh ^{n}ax\,dx=-{\frac {1}{a(n+1)}}\sinh ax\cosh ^{n+1}ax-{\frac {n+2}{n+1}}\int \cosh ^{n+2}ax\,dx\qquad {\mbox{(}}n<0{\mbox{, }}n\neq -1{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4467d1f1eb33187374543806d795720c322ee8f0)
![{\displaystyle \int {\frac {dx}{\cosh ^{n}ax}}={\frac {\sinh ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\cosh ^{n-2}ax}}\qquad {\mbox{(}}n\neq 1{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f1a4e484141e418b0092c4e5aa3570e2883b4f8)
![{\displaystyle \int x\cosh ax\,dx={\frac {1}{a}}x\sinh ax-{\frac {1}{a^{2}}}\cosh ax+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3749ec6c66bfed013d85213304dafb2594cd7e34)
![{\displaystyle \int x^{2}\cosh ax\,dx=-{\frac {2x\cosh ax}{a^{2}}}+\left({\frac {x^{2}}{a}}+{\frac {2}{a^{3}}}\right)\sinh ax+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c884996f45f9b12c1ce02d2de63ecd7136e91906)
![{\displaystyle \int \cosh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bad011eb5810859caa69e2144c833e789e03324c)
![{\displaystyle \int \cosh(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)+C\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25b712592f010239548274f2b52653eae3e594c0)
![{\displaystyle \int \cosh ax\cosh bx\,dx={\frac {1}{a^{2}-b^{2}}}(a\sinh ax\cosh bx-b\sinh bx\cosh ax)+C\qquad {\mbox{(}}a^{2}\neq b^{2}{\mbox{)}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34279bf278faa4affb647b9061654d825e50f9c3)
التكاملات الأخرى[عدل]
تكاملات دوال الظل، وظل التمام، والقاطع، وقاطع التمام الزائدية[عدل]
![{\displaystyle \int \tanh x\,dx=\ln \cosh x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d34084777ab4d5122b6fb3a917d140df1caad0dd)
![{\displaystyle \int \tanh ^{2}ax\,dx=x-{\frac {\tanh ax}{a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4100e7bd27ab3e9516e364a7f1bb1cc4f101d10e)
![{\displaystyle \int \tanh ^{n}ax\,dx=-{\frac {1}{a(n-1)}}\tanh ^{n-1}ax+\int \tanh ^{n-2}ax\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa56801fdfacb4eda9dc3fff468d055ceaa8f665)
![{\displaystyle \int \coth x\,dx=\ln |\sinh x|+C,{\text{ for }}x\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fd10aa5030fc327fc1eba419cb87581bd065282)
![{\displaystyle \int \coth ^{n}ax\,dx=-{\frac {1}{a(n-1)}}\coth ^{n-1}ax+\int \coth ^{n-2}ax\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69304e9c91b1eefcb7bc218a5080db96369e8e37)
![{\displaystyle \int \operatorname {sech} \,x\,dx=\arctan \,(\sinh x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c567185304799602087bcbe1b470a2b9e5b7880b)
![{\displaystyle \int \operatorname {csch} \,x\,dx=\ln \left|\tanh {x \over 2}\right|+C,{\text{ for }}x\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b748b884b357d180103915ef659086d9a36d4917)
التكاملات التي تحتوي على دالتي الجيب وجيب التمام الزائدية[عدل]
![{\displaystyle \int (\cosh ax)(\sinh bx)\,dx={\frac {1}{a^{2}-b^{2}}}{\big (}a(\sinh ax)(\sinh bx)-b(\cosh ax)(\cosh bx){\big )}+C\qquad {\mbox{(for }}a^{2}\neq b^{2}{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8282fbfd2c2d820938cca5be2617f08beee218f)
![{\displaystyle \int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax}}dx={\frac {\cosh ^{n-1}ax}{a(n-m)\sinh ^{m-1}ax}}+{\frac {n-1}{n-m}}\int {\frac {\cosh ^{n-2}ax}{\sinh ^{m}ax}}dx\qquad {\mbox{(for }}m\neq n{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e57829fa51b812a4655e0b9d8e7bf6d95593b8c)
أيضًا:
![{\displaystyle \int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax}}dx=-{\frac {\cosh ^{n+1}ax}{a(m-1)\sinh ^{m-1}ax}}+{\frac {n-m+2}{m-1}}\int {\frac {\cosh ^{n}ax}{\sinh ^{m-2}ax}}dx\qquad {\mbox{(for }}m\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ece99850e3bd96b507f2cef5f7bd1b1a9437a297)
![{\displaystyle \int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax}}dx=-{\frac {\cosh ^{n-1}ax}{a(m-1)\sinh ^{m-1}ax}}+{\frac {n-1}{m-1}}\int {\frac {\cosh ^{n-2}ax}{\sinh ^{m-2}ax}}dx\qquad {\mbox{(for }}m\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/344fe65522a6e9718f248d259002a03fa7018c49)
![{\displaystyle \int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax}}dx={\frac {\sinh ^{m-1}ax}{a(m-n)\cosh ^{n-1}ax}}+{\frac {m-1}{n-m}}\int {\frac {\sinh ^{m-2}ax}{\cosh ^{n}ax}}dx\qquad {\mbox{(for }}m\neq n{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6ed6639c6fc741022badcf0755565a2b4d59650)
![{\displaystyle \int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax}}dx={\frac {\sinh ^{m+1}ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {m-n+2}{n-1}}\int {\frac {\sinh ^{m}ax}{\cosh ^{n-2}ax}}dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5866475353202094f3c253511e7af93768aa3155)
![{\displaystyle \int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax}}dx=-{\frac {\sinh ^{m-1}ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {m-1}{n-1}}\int {\frac {\sinh ^{m-2}ax}{\cosh ^{n-2}ax}}dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7131fa67410f20202bfd459fdd2bf8a14aeb742)
التكاملات التي تحتوي على الدوال الزائدية والمثلثية[عدل]
![{\displaystyle \int \sinh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3caba5d0268fdc35ea5e9a9fc07b4b028ded023a)
![{\displaystyle \int \sinh(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e42067465fc2f9e4aead56de5340c0babeb9434)
![{\displaystyle \int \cosh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01cdc27e035389e4fbb28829c579b7490cd00399)
![{\displaystyle \int \cosh(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f2c51fa5c08c010c90e1aef45ac2d958f008be)
طالع أيضًا[عدل]