Wikipedia — ирекле энциклопедия проектыннан ([http://tt.wikipedia.org.ttcysuttlart1999.aylandirow.tmf.org.ru/wiki/Интеграл latin yazuında])
Риман интегралы - график астындагы кечкенә турыпочмаклар суммасына тигез
Билгеле интеграл фигура мәйданына тигез
Интеграл - интеграль хисапның төп төшенчәсе, функцияләр, саннар суммасына туры килә, билгеле интеграл функциянең графигы һәм абсцисс күчәре арасындагы мәйданга тигез, димәк кәкре сызыклы трапеция мәйданына тигез.
Ике үзгәрмә зурлык буенча интеграл табу очрагында, интеграл - функция графигының өслеге астындагы күләмгә тигез.
Риман интегралы - интеграл табу иң гади ысулы, Риман суммасы нигезендәге алым, ягъни график астындагы кечкенә турыпочмаклар суммасына тигез.
![{\displaystyle S=\sum _{i}f(x_{i})\Delta x_{i}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/617521a1703239a3ea99c335b9af665beec4ea63)
Интеграль сумма:
Бүлемнәр адымы
нульгә омтылган очракта, интеграль суммалар бертигез санга омтыла, ул интеграл дип йөртелә:
![{\displaystyle \int \limits _{a}^{b}f(x)\,dx=\lim \limits _{\delta R\to 0}\sigma _{x}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f1adafdcf21f5a2b3955495d211554ef39fd206a)
Кисемтәләр кечкенә бүлемнәрен ясап, кечкенә турыпочмаклар мәйданнары суммасы
, чик очрагында (
нульга омтылса), интеграль сумма интегралга омтыла:
![{\displaystyle S=\sum _{i}f(x_{i})\Delta x_{i}\rightarrow \int f(x)dx.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e0c7cac0106f773db4c9d554f1ba4c7d8fc93c0)
Ньютон-Лейбниц тигезләмәсе буенча баштагы функцияләр аермасы билгеле интегралга тигез:
![{\displaystyle \int k\,dx=kx+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39fba8999cf049c9826f334d9ee25938865aeb71)
![{\displaystyle \int x^{a}\,dx={\frac {x^{a+1}}{a+1}}+C\qquad {\text{(for }}a\neq -1{\text{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f79b8d0ce61ad11824db37d043d7ad888a2242c4)
![{\displaystyle \int (ax+b)^{n}\,dx={\frac {(ax+b)^{n+1}}{a(n+1)}}+C\qquad {\text{(for }}n\neq -1{\text{)}}\,\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3fcccd15d6f337c5d61126a7e23207e72c24384)
- гомуми очракта,[1]
![{\displaystyle \int {1 \over x}\,dx={\begin{cases}\ln \left|x\right|+C^{-}&x<0\\\ln \left|x\right|+C^{+}&x>0\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6714a0cb222cc0fd942b78d3e06c331f73b9e211)
![{\displaystyle \int {\frac {c}{ax+b}}\,dx={\frac {c}{a}}\ln \left|ax+b\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a2b702fce460d1ad3899bcac1720aab5f9f6406)
![{\displaystyle \int e^{ax}\,dx={\frac {1}{a}}e^{ax}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c41b04a88938a3a50de7ca4af3c567b33f9ccb37)
![{\displaystyle \int f'(x)e^{f(x)}\,dx=e^{f(x)}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91a811704bafdfdebab7ccdc39f6dedca25a1bfd)
![{\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\ln a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/544302360e06c836a24783781373750b9cb709d6)
![{\displaystyle \int \ln x\,dx=x\ln x-x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54b730b8defb012d4140543f799c9fb1a7f1a3c8)
![{\displaystyle \int \log _{a}x\,dx=x\log _{a}x-{\frac {x}{\ln a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bcd50506a3f83a358205e51f289e179c84b4d94)
![{\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|\cos {x}\right|}+C=\ln {\left|\sec {x}\right|}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fff79bf847dd0567db9119faab36c03810a868c8)
![{\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/912b48f413446f1ea54ceeec71a2f7a4f6808e42)
![{\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}}\left(x-{\frac {\sin 2x}{2}}\right)+C={\frac {1}{2}}(x-\sin x\cos x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c3915367234cede7f4f2606aacaf32b35cfcf3e)
![{\displaystyle \int \cos ^{2}x\,dx={\frac {1}{2}}\left(x+{\frac {\sin 2x}{2}}\right)+C={\frac {1}{2}}(x+\sin x\cos x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c5fd33e2ed813a3b2452c5b7bc553991b1855ea)
![{\displaystyle \int \sec ^{3}x\,dx={\frac {1}{2}}\sec x\tan x+{\frac {1}{2}}\ln |\sec x+\tan x|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e685542d54058f2defcf20dad355de10535d8d5)
![{\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/307e19c74642ddbed625e25265cb0ee59638d286)
![{\displaystyle \int \arcsin {x}\,dx=x\arcsin {x}+{\sqrt {1-x^{2}}}+C,{\text{ for }}\vert x\vert \leq +1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/574c20e5b9e7eea2e093dd394e0421d26c88e5c6)
![{\displaystyle \int \arccos {x}\,dx=x\arccos {x}-{\sqrt {1-x^{2}}}+C,{\text{ for }}\vert x\vert \leq +1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22c02f1deb70cdcb6f23af002d274d78437cae60)
![{\displaystyle \int \arctan {x}\,dx=x\arctan {x}-{\frac {1}{2}}\ln {\vert 1+x^{2}\vert }+C,{\text{ for all real }}x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2065fbb624268a059bbb9b42814a78ba22f5cc0e)
![{\displaystyle \int \operatorname {arccot} {x}\,dx=x\operatorname {arccot} {x}+{\frac {1}{2}}\ln {\vert 1+x^{2}\vert }+C,{\text{ for all real }}x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffe4f6bf1d063bcbddadcd300fb86e923e175f3b)
![{\displaystyle \int \operatorname {arcsec} {x}\,dx=x\operatorname {arcsec} {x}-\ln \vert x\,(1+{\sqrt {1-x^{-2}}}\,)\vert +C,{\text{ for }}\vert x\vert \geq +1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdf3af8e13917f0810f2ab949c6652eb4a6f2814)
![{\displaystyle \int \operatorname {arccsc} {x}\,dx=x\operatorname {arccsc} {x}+\ln \vert x\,(1+{\sqrt {1-x^{-2}}}\,)\vert +C,{\text{ for }}\vert x\vert \geq +1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df49d716aa2bbc63453c7cfc0d8f1881bd420aaf)
![{\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/d34084777ab4d5122b6fb3a917d140df1caad0dd)
![{\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 \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)
![{\displaystyle \int \operatorname {arsinh} \,x\,dx=x\,\operatorname {arsinh} \,x-{\sqrt {x^{2}+1}}+C,{\text{ for all real }}x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0dec2ea8bb791cd8cc4b41dde2c93161d1f06bf)
![{\displaystyle \int \operatorname {arcosh} \,x\,dx=x\,\operatorname {arcosh} \,x-{\sqrt {x^{2}-1}}+C,{\text{ for }}x\geq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44cc278c69df035a64634fef2e0bfb7c7a70052a)
![{\displaystyle \int \operatorname {artanh} \,x\,dx=x\,\operatorname {artanh} \,x+{\frac {\ln \left(\,1-x^{2}\right)}{2}}+C,{\text{ for }}\vert x\vert <1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24dd6a6f34722887066919ab3a475129df4c80ab)
![{\displaystyle \int \operatorname {arcoth} \,x\,dx=x\,\operatorname {arcoth} \,x+{\frac {\ln \left(x^{2}-1\right)}{2}}+C,{\text{ for }}\vert x\vert >1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a343cf5411b1fe1f49d57ae53a8785c6e8f0ad06)
![{\displaystyle \int \operatorname {arsech} \,x\,dx=x\,\operatorname {arsech} \,x+\arcsin x+C,{\text{ for }}0<x\leq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4e9040352ee7804aa8fe75bf3abb648d402c496)
![{\displaystyle \int \operatorname {arcsch} \,x\,dx=x\,\operatorname {arcsch} \,x+\vert \operatorname {arsinh} \,x\vert +C,{\text{ for }}x\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/643c6b0dce406470e1c04da2c2695691054c7e1e)
![{\displaystyle \int \cos ax\,e^{bx}\,dx={\frac {e^{bx}}{a^{2}+b^{2}}}\left(a\sin ax+b\cos ax\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46eeae4c8f9b6aeefd42dcfa860824e1d7ccb25f)
![{\displaystyle \int \sin ax\,e^{bx}\,dx={\frac {e^{bx}}{a^{2}+b^{2}}}\left(b\sin ax-a\cos ax\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/057b8b789b4e2f7a9dfc616eb57844d916e7350d)
![{\displaystyle \int \cos ax\,\cosh bx\,dx={\frac {1}{a^{2}+b^{2}}}\left(a\sin ax\,\cosh bx+b\cos ax\,\sinh bx\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe43b5be3b683620f7a38f4dffbaa9b9646cc730)
![{\displaystyle \int \sin ax\,\cosh bx\,dx={\frac {1}{a^{2}+b^{2}}}\left(b\sin ax\,\sinh bx-a\cos ax\,\cosh bx\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16b6efba234f3f63f24a1411a42aeed912cb4109)
- ↑ "Reader Survey: log|x| + C", Tom Leinster, The n-category Café, March 19, 2012