من ويكيبيديا، الموسوعة الحرة
هذه قائمة بتكاملات لمختلف الدوال في الرياضيات.[1][2]
قواعد مكاملة الدوال العامة[عدل]
![{\displaystyle \int af(x)\,dx=a\int f(x)\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56c7f8dc12de890ff42235b0e5795f07e6a9d445)
![{\displaystyle \int [f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc2c6e2acadf432d22ca42bc6a21af25e48e64d)
![{\displaystyle \int f(x)g(x)\,dx=f(x)\int g(x)\,dx-\int \left(d[f(x)]\int g(x)\,dx\right)\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2da09fdfdcb12ff1a55cb20340291b01e6832c0)
![{\displaystyle \int af(y)\,dy=a\int f(y)\,dy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff16af2bcb0230a349bf786b076798fc7b645e4)
![{\displaystyle \int [f(y)+g(y)]\,dy=\int f(y)\,dy+\int g(y)\,dy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3837cae2faae328cec936ed642265512502930a8)
![{\displaystyle \int f(y)g(y)\,dy=f(y)\int g(y)\,dy-\int \left(d[f(y)]\int g(y)\,dy\right)\,dy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0208ad5ffd2f7be5fc74eb8891970e0324d546d6)
تكاملات الدوال البسيطة[عدل]
قائمة تكاملات الدوال غير النسبية[عدل]
![{\displaystyle \int {du \over {\sqrt {a^{2}-u^{2}}}}=\arcsin {u \over a}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05ae9bd72c0b9d6455257160a0fe469763c8610a)
![{\displaystyle \int {-du \over {\sqrt {a^{2}-u^{2}}}}=\arccos {u \over a}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff8ca9ddaedbdb132b179859c79216ae795c99b5)
![{\displaystyle \int {du \over u{\sqrt {u^{2}-a^{2}}}}={1 \over a}{\mbox{arcsec}}\,{|u| \over a}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a265dea1a0e7dc6ecc4d29ef5bc1c7fe54b7404)
اللوغاريتمات[عدل]
![{\displaystyle \int \ln {x}\,dx=x\ln {x}-x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3987107bc91f514f684ac40e88852dc47e33abc9)
![{\displaystyle \int \log _{b}{x}\,dx=x\log _{b}{x}-x\log _{b}{e}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7e25004a52b2a654b87396ddf2ff3fc8a5041f9)
الدوال الأسية[عدل]
![{\displaystyle \int e^{x}\,dx=e^{x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e6d31e8ad38cc40b4e3d18ad17b756efa483abd)
![{\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\ln {a}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/816dde2034e43093b2a85e3dcc1ef2f39779f860)
![{\displaystyle \int \sin {x}\,dx=-\cos {x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/537de256cbb401203900fd3623cdbc85e31cc70b)
![{\displaystyle \int \cos {x}\,dx=\sin {x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1aae2ec756513ea8f93deb874803c61e291dd8a)
![{\displaystyle \int \tan {x}\,dx=\ln {\left|\sec {x}\right|}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/283fa05fcd07376f1e8417361ad3735c08cc3003)
![{\displaystyle \int \cot {x}\,dx=\ln {\left|\sin {x}\right|}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a79422c3c1bc1b58e8a1623920b50fb4ff87f907)
![{\displaystyle \int \sec {x}\,dx=\ln {\left|\sec {x}+\tan {x}\right|}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/378b45f5cd66c9fb7560eb362481df12ce77fa51)
![{\displaystyle \int \csc {x}\,dx=-\ln {\left|\csc {x}+\cot {x}\right|}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99de9f8865bb257f9a2fa762082e5f9d8f8ecdd0)
![{\displaystyle \int \sec ^{2}x\,dx=\tan x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f8fbfacf62d7130b7bf000e226b07f8c599bf1c)
![{\displaystyle \int \csc ^{2}x\,dx=-\cot x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/364c3afec409bb6bfbb787276d7cfd884040b07a)
![{\displaystyle \int \sec {x}\,\tan {x}\,dx=\sec {x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/385d180bf75e276f8b0cafb1fdc1f584554be54f)
![{\displaystyle \int \csc {x}\,\cot {x}\,dx=-\csc {x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/038c3e132b5c6826b7be055d24fa617842c493d2)
![{\displaystyle \int \sin ^{2}x\,dx={\frac {1}{2}}(x-\sin x\cos x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f51b216b7c2533bf9ceb78b278f162070ccd2f9)
![{\displaystyle \int \cos ^{2}x\,dx={\frac {1}{2}}(x+\sin x\cos x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f423c27f97e0cac690bdb3e0faf09dea37772362)
![{\displaystyle \int \sin ^{n}x\,dx=-{\frac {\sin ^{n-1}{x}\cos {x}}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}{x}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c15d560c4a9f07da5aa62b1adc435b6e785ea33)
![{\displaystyle \int \cos ^{n}x\,dx=-{\frac {\cos ^{n-1}{x}\sin {x}}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}{x}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3aabda4cb3227cd25ac6399175cb752b3a284e8)
![{\displaystyle \int \arctan {x}\,dx=x\,\arctan {x}-{\frac {1}{2}}\ln {\left|1+x^{2}\right|}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cf3499100d93064704955a0addfbfaf2f9726b1)
دوال القطع الزائد[عدل]
![{\displaystyle \int \sinh x\,dx=\cosh x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a452d5b48cae9335f0a79d19b85a61d28154683a)
![{\displaystyle \int \cosh x\,dx=\sinh x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/529344aa89d4a7732c58734fa5134612b73aaa19)
![{\displaystyle \int \tanh x\,dx=\ln |\cosh x|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e448d5b96361c98f3d0af72e0a9e860261bfe9d4)
![{\displaystyle \int {\mbox{csch}}\,x\,dx=\ln \left|\tanh {x \over 2}\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53e784b82203b2db8d3bb9435d677aa204705ef1)
![{\displaystyle \int {\mbox{sech}}\,x\,dx=\arctan(\sinh x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0cba1e31fd35b44ba1cd78d5ec48f68be1d5f7a8)
![{\displaystyle \int \coth x\,dx=\ln |\sinh x|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd7bd1bfe08e160d8d488e245bd13c42a16c91d)
تكاملات محددة[عدل]
(أنظر أيضا دالة غاما)
![{\displaystyle \int _{0}^{\infty }{e^{-x^{2}}\,dx}={\frac {1}{2}}{\sqrt {\pi }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16b05b4bc20fb5de48a721eed507e9f61580d0b3)
(أنظر أيضا عدد بيرنولي)
![{\displaystyle \int _{0}^{\infty }{{\frac {x^{3}}{e^{x}-1}}\,dx}={\frac {\pi ^{4}}{15}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/493906f6230047e8a8a156eccbafa743875182ff)
![{\displaystyle \int _{0}^{\infty }{\frac {\sin(x)}{x}}\,dx={\frac {\pi }{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a1f77f796f0a822c23b3254bf028f90d7570509)
(حيث
هي دالة غاما.)
![{\displaystyle \int _{-\infty }^{\infty }e^{-(ax^{2}+bx+c)}\,dx={\sqrt {\frac {\pi }{a}}}e^{\frac {b^{2}-4ac}{4a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89808d6ed9544906bbe91a180cd5d5fae5ae2177)
مراجع[عدل]