«Википеди» ирĕклĕ энциклопединчи материал
Аяларах иррационаллă функцисен интегралĕсен (умсăнарĕсен) йышне илсе кăтартнă. Пур çĕрте те аддитивлă констаттăна катертнĕ.
Аяларах пур çĕрте те:
.
![{\displaystyle \int r\;dx={\frac {1}{2}}\left(xr+a^{2}\,\ln \left({x+r}\right)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fec03963458793ecc23d45ad71a82014ec08bc53)
![{\displaystyle \int r^{3}\;dx={\frac {1}{4}}xr^{3}+{\frac {1}{8}}3a^{2}xr+{\frac {3}{8}}a^{4}\ln \left({\frac {x+r}{a}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94dfed8d5f78aae70502037ff46d16cafa21f835)
![{\displaystyle \int r^{5}\;dx={\frac {1}{6}}xr^{5}+{\frac {5}{24}}a^{2}xr^{3}+{\frac {5}{16}}a^{4}xr+{\frac {5}{16}}a^{6}\ln \left({\frac {x+r}{a}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b509300a32a725ea96136424c71aee30a99de89)
![{\displaystyle \int xr^{2n+1}\;dx={\frac {r^{2n+3}}{2n+3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6583ead420aa74655131aaf0ed82cbb82e36157)
![{\displaystyle \int x^{2}r\;dx={\frac {xr^{3}}{4}}-{\frac {a^{2}xr}{8}}-{\frac {a^{4}}{8}}\ln \left({\frac {x+r}{a}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be3188b42d7d35dd388544a23422b32a0d11aa54)
![{\displaystyle \int x^{2}r^{3}\;dx={\frac {xr^{5}}{6}}-{\frac {a^{2}xr^{3}}{24}}-{\frac {a^{4}xr}{16}}-{\frac {a^{6}}{16}}\ln \left({\frac {x+r}{a}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44c78e6b370f74920f2f62974e81ca1da7de0687)
![{\displaystyle \int x^{3}r\;dx={\frac {r^{5}}{5}}-{\frac {a^{2}r^{3}}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c286c93cf9b39345e6c65e118526cf6b0ebba341)
![{\displaystyle \int x^{3}r^{3}\;dx={\frac {r^{7}}{7}}-{\frac {a^{2}r^{5}}{5}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ca1e44ac2d54d893799548feac05e8684a650d4)
![{\displaystyle \int x^{3}r^{2n+1}\;dx={\frac {r^{2n+5}}{2n+5}}-{\frac {a^{3}r^{2n+3}}{2n+3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f48c6fc659e5a2eb4a86664f238b542283bc1ee2)
![{\displaystyle \int x^{4}r\;dx={\frac {x^{3}r^{3}}{6}}-{\frac {a^{2}xr^{3}}{8}}+{\frac {a^{4}xr}{16}}+{\frac {a^{6}}{16}}\ln \left({\frac {x+r}{a}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd52a5e88ef2b63a48b0cf31a0a2955ac9d42cc6)
![{\displaystyle \int x^{4}r^{3}\;dx={\frac {x^{3}r^{5}}{8}}-{\frac {a^{2}xr^{5}}{16}}+{\frac {a^{4}xr^{3}}{64}}+{\frac {3a^{6}xr}{128}}+{\frac {3a^{8}}{128}}\ln \left({\frac {x+r}{a}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea30e9a53a9c7d2b93d58ff4532c6765e3ce9196)
![{\displaystyle \int x^{5}r\;dx={\frac {r^{7}}{7}}-{\frac {2a^{2}r^{5}}{5}}+{\frac {a^{4}r^{3}}{3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cbd4bfbe0c7d57a09ac1901c6522cb7e8327d48)
![{\displaystyle \int x^{5}r^{3}\;dx={\frac {r^{9}}{9}}-{\frac {2a^{2}r^{7}}{7}}+{\frac {a^{4}r^{5}}{5}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/982e3258c17a21f19d9bdfa66ccbb2e675a32564)
![{\displaystyle \int x^{5}r^{2n+1}\;dx={\frac {r^{2n+7}}{2n+7}}-{\frac {2a^{2}r^{2n+5}}{2n+5}}+{\frac {a^{4}r^{2n+3}}{2n+3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e37516c175afbb2ba492a368426ca0a68db63935)
![{\displaystyle \int {\frac {r\;dx}{x}}=r-a\ln \left|{\frac {a+r}{x}}\right|=r-a\operatorname {arsh} {\frac {a}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6982563bf3c4f66008771f8b42f8bf10bd74bed5)
![{\displaystyle \int {\frac {r^{3}\;dx}{x}}={\frac {r^{3}}{3}}+a^{2}r-a^{3}\ln \left|{\frac {a+r}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a99fe55679d157911cd614e9ca510e54d729d42)
![{\displaystyle \int {\frac {r^{5}\;dx}{x}}={\frac {r^{5}}{5}}+{\frac {a^{2}r^{3}}{3}}+a^{4}r-a^{5}\ln \left|{\frac {a+r}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d35f2538f5e4bbde4837e0642fdceb6e151c5062)
![{\displaystyle \int {\frac {r^{7}\;dx}{x}}={\frac {r^{7}}{7}}+{\frac {a^{2}r^{5}}{5}}+{\frac {a^{4}r^{3}}{3}}+a^{6}r-a^{7}\ln \left|{\frac {a+r}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0505fdb3e13b7e0704f61b5285649e041ec8d66)
![{\displaystyle \int {\frac {dx}{r}}=\operatorname {arsh} {\frac {x}{a}}=\ln \left|x+r\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90eeb213055360ae0184ce93b6e021f101ac699f)
![{\displaystyle \int {\frac {x\,dx}{r}}=r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77a3156a33834a19e192af67f3a54da6ddbe4ce1)
![{\displaystyle \int {\frac {x^{2}\;dx}{r}}={\frac {x}{2}}r-{\frac {a^{2}}{2}}\,\operatorname {arsh} {\frac {x}{a}}={\frac {x}{2}}r-{\frac {a^{2}}{2}}\ln \left|x+r\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70046ed04208318c08fe6cad4112ff484a24c58e)
![{\displaystyle \int {\frac {dx}{xr}}=-{\frac {1}{a}}\,\operatorname {arsh} {\frac {a}{x}}=-{\frac {1}{a}}\ln \left|{\frac {a+r}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d8fd0ac7916742ff9539e66dbd7d95f0c3ebded)
Аяларах пур тĕлте те:
.
Йышăннă:
; енчен те
пулсан, тепĕр секцие пăхăр.
![{\displaystyle \int s\;dx={\frac {1}{2}}\left(xs-a^{2}\ln \left(x+s\right)\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e4e62e9f5373e4e492a8da396f711c4b259518d)
![{\displaystyle \int xs\;dx=-{\frac {1}{3}}s^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8c5d60987f74786f543fc97e76c98c18c604780)
![{\displaystyle \int {\frac {s\;dx}{x}}=s-a\cos ^{-1}\left|{\frac {a}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e3796397e7bf4a7e7654dd17d20474cf8bdec98)
![{\displaystyle \int {\frac {dx}{s}}=\int {\frac {dx}{\sqrt {x^{2}-a^{2}}}}=\ln \left|x+s\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d079ad25c933c64cc71a381f6bea6dc254bd899c)
Асăрхăпăр, что
, где
принимает только положительные значения.
![{\displaystyle \int {\frac {x\;dx}{s}}=s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c17011581cfa3d9e53920f319ac77a75013dec77)
![{\displaystyle \int {\frac {x\;dx}{s^{3}}}=-{\frac {1}{s}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0f2035386db8b8331fafbbb54ba1c1cf994fb5d)
![{\displaystyle \int {\frac {x\;dx}{s^{5}}}=-{\frac {1}{3s^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bcfb12b7156194900353c9def5d1e095149ce99)
![{\displaystyle \int {\frac {x\;dx}{s^{7}}}=-{\frac {1}{5s^{5}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7db234e0dc43716a52f17f53c9b79e30303aaf18)
![{\displaystyle \int {\frac {x\;dx}{s^{2n+1}}}=-{\frac {1}{(2n-1)s^{2n-1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/808ffdc762b49c281071fd22b77938afa3e0cd5a)
![{\displaystyle \int {\frac {x^{2m}\;dx}{s^{2n+1}}}=-{\frac {1}{2n-1}}{\frac {x^{2m-1}}{s^{2n-1}}}+{\frac {2m-1}{2n-1}}\int {\frac {x^{2m-2}\;dx}{s^{2n-1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/819c367c3f01849bfcfa4f310a2e2fc8d4630d2e)
![{\displaystyle \int {\frac {x^{2}\;dx}{s}}={\frac {xs}{2}}+{\frac {a^{2}}{2}}\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25191a1c6fa21c48b6b89f476b96e8c7fa54a541)
![{\displaystyle \int {\frac {x^{2}\;dx}{s^{3}}}=-{\frac {x}{s}}+\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/755124bfe54faeb5f7029ee29993b42e0335da23)
![{\displaystyle \int {\frac {x^{4}\;dx}{s}}={\frac {x^{3}s}{4}}+{\frac {3}{8}}a^{2}xs+{\frac {3}{8}}a^{4}\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95143cd1e023e990284ba780dcf4d8d6089ac254)
![{\displaystyle \int {\frac {x^{4}\;dx}{s^{3}}}={\frac {xs}{2}}-{\frac {a^{2}x}{s}}+{\frac {3}{2}}a^{2}\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1ba248b7ebdb39ad25a8b8e98996b7b7994162e)
![{\displaystyle \int {\frac {x^{4}\;dx}{s^{5}}}=-{\frac {x}{s}}-{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}+\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88d29db4a76fe21b51b57e5e47e8136b88280fd5)
где ![{\displaystyle n>m\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d51181badc60126c983261031d3ac9544eac5008)
![{\displaystyle \int {\frac {dx}{s^{3}}}=-{\frac {1}{a^{2}}}{\frac {x}{s}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/676ac2308ed60218f4e246884e5783df8e2ebc54)
![{\displaystyle \int {\frac {dx}{s^{5}}}={\frac {1}{a^{4}}}\left[{\frac {x}{s}}-{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/054a5959ce5e03cf279c1b29dff2ba014ac6dcde)
![{\displaystyle \int {\frac {dx}{s^{7}}}=-{\frac {1}{a^{6}}}\left[{\frac {x}{s}}-{\frac {2}{3}}{\frac {x^{3}}{s^{3}}}+{\frac {1}{5}}{\frac {x^{5}}{s^{5}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86843311de7fc72bc01f87742445f7c4b88899e9)
![{\displaystyle \int {\frac {dx}{s^{9}}}={\frac {1}{a^{8}}}\left[{\frac {x}{s}}-{\frac {3}{3}}{\frac {x^{3}}{s^{3}}}+{\frac {3}{5}}{\frac {x^{5}}{s^{5}}}-{\frac {1}{7}}{\frac {x^{7}}{s^{7}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ca32b3a8d7f9040840f5d1de3467129edff0d80b)
![{\displaystyle \int {\frac {x^{2}\;dx}{s^{5}}}=-{\frac {1}{a^{2}}}{\frac {x^{3}}{3s^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d0c92cdcb44ecfe7179711341a1964ef2a0782f)
![{\displaystyle \int {\frac {x^{2}\;dx}{s^{7}}}={\frac {1}{a^{4}}}\left[{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}-{\frac {1}{5}}{\frac {x^{5}}{s^{5}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96ea4b7b2973dd3e2affa09931a8bf41316161f1)
![{\displaystyle \int {\frac {x^{2}\;dx}{s^{9}}}=-{\frac {1}{a^{6}}}\left[{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}-{\frac {2}{5}}{\frac {x^{5}}{s^{5}}}+{\frac {1}{7}}{\frac {x^{7}}{s^{7}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/239ff6c41a3342440c712b9f0c4940e8e6a000d2)
Аяларах пур тĕлте те:
![{\displaystyle \int t\;dx={\frac {1}{2}}\left(xt+a^{2}\arcsin {\frac {x}{a}}\right)={\frac {1}{2}}\left(xt-a^{2}\arccos {\frac {x}{a}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57de59b3df7e4db6d7a2ec591d41d578bbc1541d)
![{\displaystyle \int xt\;dx=-{\frac {1}{3}}t^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27a7955f48d65a6b71701ac91b482d3cd52135d9)
![{\displaystyle \int {\frac {t\;dx}{x}}=t-a\ln \left|{\frac {a+t}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5640e01ca93f088bad3796601d29df0447389e49)
![{\displaystyle \int {\frac {dx}{t}}=\arcsin {\frac {x}{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d77ab7a32234bae3f727df2f9def79d3574b4532)
![{\displaystyle \int {\frac {x\;dx}{t}}=-t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d130b6aa729a4790b54608a8cb7c91683d3893b)
![{\displaystyle \int {\frac {x^{2}\;dx}{t}}=-{\frac {x}{2}}t+{\frac {a^{2}}{2}}\arcsin {\frac {x}{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db44e7f02cc206b7cd4645cfcd5c3e356cf0bdc7)
![{\displaystyle \int t\;dx={\frac {1}{2}}\left(xt-\operatorname {sgn} x\,\operatorname {arch} \left|{\frac {x}{a}}\right|\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56141d3a132569370676a5468f1f76af77016547)
Кунта:
![{\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\ln \left|2{\sqrt {aR}}+2ax+b\right|\qquad {\mbox{( }}a>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b96a33662681c77d8151a6a7216f3c3533d0fa94)
![{\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\,\operatorname {arsh} {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{( }}a>0{\mbox{, }}4ac-b^{2}>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da63fa0f8ca66e48ade8380319db7c0cc76bf15a)
![{\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\ln |2ax+b|\quad {\mbox{( }}a>0{\mbox{, }}4ac-b^{2}=0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d71e228f4d72d4940b70a3d145ddbd0a4872d3e3)
![{\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}=-{\frac {1}{\sqrt {-a}}}\arcsin {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad {\mbox{( }}a<0{\mbox{, }}4ac-b^{2}<0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b9c68c703c24d04d413fda75dc8ead3361eef75)
![{\displaystyle \int {\frac {dx}{\sqrt {(ax^{2}+bx+c)^{3}}}}={\frac {4ax+2b}{(4ac-b^{2}){\sqrt {R}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/350bd207dd27cf5a13d9cedd433d471f6bf5b10a)
![{\displaystyle \int {\frac {dx}{\sqrt {(ax^{2}+bx+c)^{5}}}}={\frac {4ax+2b}{3(4ac-b^{2}){\sqrt {R}}}}\left({\frac {1}{R}}+{\frac {8a}{4ac-b^{2}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e98cfb50fe31a8935b8700784450dedb983e8b0)
![{\displaystyle \int {\frac {dx}{\sqrt {(ax^{2}+bx+c)^{2n+1}}}}={\frac {4ax+2b}{(2n-1)(4ac-b^{2})R^{(2n-1)/2}}}+{\frac {8a(n-1)}{(2n-1)(4ac-b^{2})}}\int {\frac {dx}{R^{(2n-1)/2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e917fd2f53afd23bbd44fef4d625ea736c85ff2c)
![{\displaystyle \int {\frac {x\;dx}{\sqrt {ax^{2}+bx+c}}}={\frac {\sqrt {R}}{a}}-{\frac {b}{2a}}\int {\frac {dx}{\sqrt {R}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/600030a37c84403934202860a7eb9ae3b316cd92)
![{\displaystyle \int {\frac {x\;dx}{\sqrt {(ax^{2}+bx+c)^{3}}}}=-{\frac {2bx+4c}{(4ac-b^{2}){\sqrt {R}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ac28c9be68dc63f9d05f11b71183336d6997a31)
![{\displaystyle \int {\frac {x\;dx}{\sqrt {(ax^{2}+bx+c)^{2n+1}}}}=-{\frac {1}{(2n-1)aR^{(2n-1)/2}}}-{\frac {b}{2a}}\int {\frac {dx}{R^{(2n+1)/2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37e4359d43619c72193e0d13da99da1e0ac6d7cf)
![{\displaystyle \int {\frac {dx}{x{\sqrt {ax^{2}+bx+c}}}}=-{\frac {1}{\sqrt {c}}}\ln \left({\frac {2{\sqrt {cR}}+bx+2c}{x}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/25d4bed88b8f638349e32532e1ae1d7917ef85d1)
![{\displaystyle \int {\frac {dx}{x{\sqrt {ax^{2}+bx+c}}}}=-{\frac {1}{\sqrt {c}}}\operatorname {arsh} \left({\frac {bx+2c}{|x|{\sqrt {4ac-b^{2}}}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bc5f151f1d80f8ea8d04e04efa574254f85ef06)
, где последний интеграл находится в зависимости от параметров a,b и c (см. выше)
![{\displaystyle \int {\frac {dx}{x{\sqrt {ax+b}}}}\,=\,{\frac {-2}{\sqrt {b}}}\operatorname {arth} {\sqrt {\frac {ax+b}{b}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/254f69157c8579ea7e8955789455a178ca6ef807)
![{\displaystyle \int {\frac {\sqrt {ax+b}}{x}}\,dx\;=\;2\left({\sqrt {ax+b}}-{\sqrt {b}}\operatorname {arth} {\sqrt {\frac {ax+b}{b}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8be1058c0204e6c4ba6360ff4f756a250cacf66)
![{\displaystyle \int {\frac {x^{n}}{\sqrt {ax+b}}}\,dx\;=\;{\frac {2}{a(2n+1)}}\left(x^{n}{\sqrt {ax+b}}-bn\int {\frac {x^{n-1}}{\sqrt {ax+b}}}\,dx\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc428e49a57674b857b5c7373aa652a042179e60)
![{\displaystyle \int x^{n}{\sqrt {ax+b}}\,dx\;=\;{\frac {2}{2n+1}}\left(x^{n+1}{\sqrt {ax+b}}+bx^{n}{\sqrt {ax+b}}-nb\int x^{n-1}{\sqrt {ax+b}}\,dx\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd8c3bf8206cec454702eaab4cc9d3bf911983b1)
- Кĕнекесем
- Градштейн И. С. Рыжик И. М. Таблицы интегралов, сумм, рядов и произведений. — 4-е изд. — М.: Наука, 1963. — ISBN 0-12-294757-6 // EqWorld
- Двайт Г. Б. Таблицы интегралов СПб: Издательство и типография АО ВНИИГ им. Б. В. Веденеева, 1995. — 176 с. — ISBN 5-85529-029-8.
- D. Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st ed., 2002. ISBN 1-58488-291-3.
- M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 1964. ISBN 0-486-61272-4
- Интегралсен таблицисем
- Интегралсене шутлани