出典: フリー百科事典『ウィキペディア(Wikipedia)』
| この記事は検証可能な参考文献や出典が全く示されていないか、不十分です。出典を追加して記事の信頼性向上にご協力ください。(このテンプレートの使い方) 出典検索?: "原始関数の一覧" – ニュース · 書籍 · スカラー · CiNii · J-STAGE · NDL · dlib.jp · ジャパンサーチ · TWL(2016年1月) |
本項は、原始関数の一覧(げんしかんすうのいちらん)である。以下、積分定数は
とする。
を含む積分[編集]
![{\displaystyle \int {\frac {1}{ax+b}}\,dx={\frac {1}{a}}\ln \left|ax+b\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7d3d9c89a3b5207efed294f709ece99ec983260)
![{\displaystyle \int {\frac {x}{ax+b}}\,dx={\frac {x}{a}}-{\frac {b}{a^{2}}}\ln \left|ax+b\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0044cf33077ed2838244a1f49755bfda52d1b5c2)
![{\displaystyle \int {\frac {x^{2}}{ax+b}}\,dx={\frac {1}{2a^{3}}}(a^{2}x^{2}-2abx+2b^{2}\ln \left|ax+b\right|)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7c7c60488737cc07dddb3a3e3e83c515a5b526c)
![{\displaystyle \int {\frac {1}{x(ax+b)}}\,dx=-{\frac {1}{b}}\ln \left|{\frac {ax+b}{x}}\right|+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0eb20d2f8f53fa28110b76a629615131429bbdea)
![{\displaystyle \int {\frac {1}{x^{2}(ax+b)}}\,dx={\frac {a}{b^{2}}}\ln \left|{\frac {ax+b}{x}}\right|-{\frac {1}{bx}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c659a46704fdcb248a22f4dc0900bc8eef90bdc)
を含む積分[編集]
![{\displaystyle \int x{\sqrt {a+bx}}\,dx={\frac {2}{15b^{2}}}(3bx-2a)(a+bx)^{\frac {3}{2}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b524730cf4c9202a9be0b471eb9fb10a571077eb)
![{\displaystyle \int x^{2}{\sqrt {a+bx}}\,dx={\frac {2}{105b^{3}}}(15b^{2}x^{2}-12abx+8a^{2})(a+bx)^{\frac {3}{2}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a62d6f9803252581082eae5b5fd6b48f91b13389)
![{\displaystyle \int x^{n}{\sqrt {a+bx}}\,dx={\frac {2}{b(2n+3)}}x^{n}(a+bx)^{\frac {3}{2}}-{\frac {2na}{b(2n+3)}}\int x^{n-1}{\sqrt {a+bx}}dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/970ecba27d457e5e85a6fcfa9875c9873f2193aa)
![{\displaystyle \int {\frac {\sqrt {a+bx}}{x}}\,dx=2{\sqrt {a+bx}}+a\int {\frac {1}{x{\sqrt {a+bx}}}}dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af4438c94f5b26aa88772dc7cff1b9949dc805b2)
![{\displaystyle \int {\frac {\sqrt {a+bx}}{x^{n}}}\,dx={\frac {-1}{a(n-1)}}{\frac {(a+bx)^{\frac {3}{2}}}{x^{n-1}}}-{\frac {(2n-5)b}{2a(n-1)}}\int {\frac {\sqrt {a+bx}}{x^{n-1}}}dx,n\neq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/757451d97cad99ceb19536abcb8257de452776c2)
![{\displaystyle ={\frac {2}{\sqrt {-a}}}\arctan {\sqrt {\frac {a+bx}{-a}}}+C,a<0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/514434d7d71604ad00cccb401c2beb5c537a69fb)
![{\displaystyle \int {\frac {1}{x^{n}{\sqrt {a+bx}}}}\,dx={\frac {-1}{a(n-1)}}{\frac {\sqrt {a+bx}}{x^{n-1}}}-{\frac {(2n-3)b}{2a(n-1)}}\int {\frac {1}{x^{n-1}}}{\sqrt {a+bx}}dx,n\neq 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3346722db934b4aae8ac5988a242ba6d7dcba87a)
を含む積分[編集]
![{\displaystyle \int {\frac {1}{x^{2}+\alpha ^{2}}}\,dx={\frac {1}{\alpha }}\arctan {\frac {x}{\alpha }}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71c7e018fe6db9277a40fdfd7982cd1190ef6e49)
![{\displaystyle \int {\frac {1}{\pm x^{2}\mp \alpha ^{2}}}\,dx={\frac {1}{2\alpha }}\ln \left({\dfrac {x\mp \alpha }{\pm x+\alpha }}\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52ff61be1e2251c388343f7628a065cd0c18cf68)
を含む積分[編集]
![{\displaystyle \int {\frac {1}{ax^{2}+b}}\,dx={\frac {1}{\sqrt {ab}}}\arctan {\sqrt {\frac {a}{b}}}x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5e92ad441f3f6cf025726cfcab3e63129315497)
を含む積分[編集]
![{\displaystyle \int (ax^{2}+bx+c)\,dx={\frac {ax^{3}}{3}}+{\frac {bx^{2}}{2}}+cx+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47d76d95aae7f89080ecda8273fb196084f68db0)
を含む積分[編集]
![{\displaystyle \int {\sqrt {a^{2}+x^{2}}}\,dx={\frac {1}{2}}x{\sqrt {a^{2}+x^{2}}}+{\frac {1}{2}}a^{2}\ln \left(x+{\sqrt {a^{2}+x^{2}}}\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb69f95bcc631b60d11a5703ce9d7af195736e65)
![{\displaystyle \int x^{2}{\sqrt {a^{2}+x^{2}}}\,dx={\frac {1}{8}}x(a^{2}+2x^{2}){\sqrt {a^{2}+x^{2}}}-{\frac {1}{8}}a^{4}\ln \left(x+{\sqrt {a^{2}+x^{2}}}\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21f116debbc99e9b70c5ea79b621652ce59a8bd6)
![{\displaystyle \int {\frac {\sqrt {a^{2}+x^{2}}}{x}}\,dx={\sqrt {a^{2}+x^{2}}}-a\ln \left({\frac {a+{\sqrt {a^{2}+x^{2}}}}{x}}\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a44dcbbc58c2a2c86a4d396644b371d989b1cdc)
![{\displaystyle \int {\frac {\sqrt {a^{2}+x^{2}}}{x^{2}}}\,dx=\ln \left(x+{\sqrt {a^{2}+x^{2}}}\right)-{\frac {\sqrt {a^{2}+x^{2}}}{x}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad76e7cb7cf9581f08c9e1854db9be18c1cabe56)
![{\displaystyle \int {\frac {1}{\sqrt {a^{2}+x^{2}}}}\,dx=\ln \left(x+{\sqrt {a^{2}+x^{2}}}\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f817cee6625a9004402de071a3d25b6df14bfd8b)
![{\displaystyle \int {\frac {x^{2}}{\sqrt {a^{2}+x^{2}}}}\,dx={\frac {1}{2}}x{\sqrt {a^{2}+x^{2}}}-{\frac {1}{2}}a^{2}\ln \left({\sqrt {a^{2}+x^{2}}}+x\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/805ad32f939114f0c3c61e04f54f87841074c1f6)
![{\displaystyle \int {\frac {1}{x{\sqrt {a^{2}+x^{2}}}}}\,dx={\frac {1}{a}}\ln \left({\frac {x}{a+{\sqrt {a^{2}+x^{2}}}}}\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05c2243f6d7c6bfc6850460d13b900a7b28b9009)
![{\displaystyle \int {\frac {1}{x^{2}{\sqrt {a^{2}+x^{2}}}}}\,dx=-{\frac {\sqrt {a^{2}+x^{2}}}{a^{2}x}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07db5aaec82568272c1bafa148f55b7e3c7b5d09)
を含む積分[編集]
![{\displaystyle \int {\frac {1}{\sqrt {x^{2}-a^{2}}}}\,dx=\ln \left(x+{\sqrt {x^{2}-a^{2}}}\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a827e838850bf7e0068ba15a505a78527337101)
を含む積分[編集]
![{\displaystyle \int {\frac {1}{\sqrt {a^{2}-x^{2}}}}\,dx=\arcsin {\frac {x}{a}}+C=-\arccos {\frac {x}{a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d48328c9876dbfc8d2b959559f8e34e312d0354e)
![{\displaystyle \int {\sqrt {a^{2}-x^{2}}}\,dx={\frac {1}{2}}x{\sqrt {a^{2}-x^{2}}}+{\frac {a^{2}}{2}}\arcsin {\frac {x}{a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93920ca55fca9d71ffc2097b3813dea6ac51fe0b)
![{\displaystyle \int x^{2}{\sqrt {a^{2}-x^{2}}}\,dx={\frac {1}{8}}x(2x^{2}-a^{2}){\sqrt {a^{2}-x^{2}}}+{\frac {1}{8}}a^{4}\arcsin {\frac {x}{a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/262f41d8ef5af119baeb878482da18d2cc004260)
![{\displaystyle \int {\frac {\sqrt {a^{2}-x^{2}}}{x}}\,dx={\sqrt {a^{2}-x^{2}}}-a\ln \left({\frac {a+{\sqrt {a^{2}-x^{2}}}}{x}}\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8a272ad729e21179ce2b79aca9e2644da7cd015)
![{\displaystyle \int {\frac {\sqrt {a^{2}-x^{2}}}{x^{2}}}\,dx=-{\frac {\sqrt {a^{2}-x^{2}}}{x}}-\arcsin {\frac {x}{a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d7fdbe0760b4f876ea88b868dc02c236b45247b)
![{\displaystyle \int {\frac {1}{x{\sqrt {a^{2}-x^{2}}}}}\,dx=-{\frac {1}{a}}\ln \left({\frac {a+{\sqrt {a^{2}-x^{2}}}}{x}}\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/697e36fd103ef2dc55a69e20dba6f026e4969925)
![{\displaystyle \int {\frac {x^{2}}{\sqrt {a^{2}-x^{2}}}}\,dx=-{\frac {1}{2}}x{\sqrt {a^{2}-x^{2}}}+{\frac {1}{2}}a^{2}\arcsin {\frac {x}{a}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20ebaca5cb32dbe082e985d92b3c4227229c5049)
![{\displaystyle \int {\frac {1}{x^{2}{\sqrt {a^{2}-x^{2}}}}}\,dx=-{\frac {\sqrt {a^{2}-x^{2}}}{a^{2}x}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a595bbf2c10ae6304ffba5056b4dc177af32c4b)
を含む積分[編集]
![{\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\ln \left(2{\sqrt {a}}R+2ax+b\right)\qquad ({\mbox{for }}a>0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea8ea1d75eb23bf9bf280b5b183fc295f726fb92)
![{\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\,\operatorname {arsinh} {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{(for }}a>0{\mbox{, }}4ac-b^{2}>0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/837e1ab91fe899e88b6f3c4b13666e8697eb3013)
![{\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\ln |2ax+b|\quad {\mbox{(for }}a>0{\mbox{, }}4ac-b^{2}=0{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/556efdcbf8ee92bfb28e48482149c769d9125052)
![{\displaystyle \int {\frac {dx}{R}}=-{\frac {1}{\sqrt {-a}}}\arcsin {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad {\mbox{(for }}a<0{\mbox{, }}4ac-b^{2}<0{\mbox{, }}\left(2ax+b\right)<{\sqrt {b^{2}-4ac}}{\mbox{)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ff43523d8ced40e41a8295125d0ddfcf66ada9c)
![{\displaystyle \int {\frac {dx}{R^{3}}}={\frac {4ax+2b}{(4ac-b^{2})R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/086934b294e8b53bebe7b53241bad912f4212dee)
![{\displaystyle \int {\frac {dx}{R^{5}}}={\frac {4ax+2b}{3(4ac-b^{2})R}}\left({\frac {1}{R^{2}}}+{\frac {8a}{4ac-b^{2}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6887eff55e44af7ed031fa1d919d3de3f379a90b)
![{\displaystyle \int {\frac {dx}{R^{2n+1}}}={\frac {2}{(2n-1)(4ac-b^{2})}}\left({\frac {2ax+b}{R^{2n-1}}}+4a(n-1)\int {\frac {dx}{R^{2n-1}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6fd19c82abfd6ab01d93cc3f2691059d4b4915c)
![{\displaystyle \int {\frac {x}{R}}\;dx={\frac {R}{a}}-{\frac {b}{2a}}\int {\frac {dx}{R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6880a823009f7d8fe7e2579fa228324ef0d15f03)
![{\displaystyle \int {\frac {x}{R^{3}}}\;dx=-{\frac {2bx+4c}{(4ac-b^{2})R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96545813e77cd2cc46ac4034b607a9457df212a0)
![{\displaystyle \int {\frac {x}{R^{2n+1}}}\;dx=-{\frac {1}{(2n-1)aR^{2n-1}}}-{\frac {b}{2a}}\int {\frac {dx}{R^{2n+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de6de2ef912d111ee50377af9daaf84b45633498)
![{\displaystyle \int {\frac {dx}{xR}}=-{\frac {1}{\sqrt {c}}}\ln \left({\frac {2{\sqrt {c}}R+bx+2c}{x}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c4c2981ae219f385ff1e91f03659b9d15b9a836)
![{\displaystyle \int {\frac {dx}{xR}}=-{\frac {1}{\sqrt {c}}}\operatorname {arsinh} \left({\frac {bx+2c}{|x|{\sqrt {4ac-b^{2}}}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa9d34b7cd7ee7609559c1c8e42a3c032594fb4)
三角関数を含む積分[編集]
![{\displaystyle \int \cos x\,dx=\sin x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97f4ac900d091510d57526b583655614b90abb40)
![{\displaystyle \int -\sin x\,dx=\cos x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d071072f6c67bc02b079a8cd5eca69a6454e052)
![{\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/417ebf76db7dc32be8faeb5c7e845ae497c68389)
![{\displaystyle \int \sec x\tan x\,dx=\sec x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c423ff0ccab623b12516ca09c517bab832afcab)
![{\displaystyle \int -\csc x\cot x\,dx=\csc x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c94476d9f69be8d1ddc94d07335d876f66af76c5)
![{\displaystyle \int \tan x\,dx=-\ln(\cos x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73430027d96f7d9e834d475b6172e7256dbefcec)
![{\displaystyle \int \cot x\,dx=\ln(\sin x)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e651266eca901ddc4169372d29e2fa3ac3a02693)
:グーデルマン関数の逆関数
![{\displaystyle \int \csc x\,dx=-\ln(\csc x+\cot x)+C=\ln \left({\tan x-\sin x \over \sin x\tan x}\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d72d773b7231ed1cf895931c409be681eddc4aaf)
![{\displaystyle \int \sin ^{n}x\,dx=-{\frac {1}{n}}\sin ^{n-1}x\cos x+{\frac {n-1}{n}}\int \sin ^{n-2}x\,dx+C\quad \forall n\geq 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f72c64755ec77905174dc1cbbb908d2cc603549b)
![{\displaystyle \int \sin ^{2}x\,dx={\frac {x}{2}}-{\frac {\sin {2x}}{4}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aac271e8ee4ab84707f3f5d91dc3b53dc256834)
![{\displaystyle \int \cos ^{n}x\,dx={\frac {1}{n}}\cos ^{n-1}x\sin x+{\frac {n-1}{n}}\int \cos ^{n-2}x\,dx+C\quad \forall n\geq 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab0fb2de7e529c5d4e187f74c75151fb34bed0b9)
![{\displaystyle \int \cos ^{2}x\,dx={\frac {x}{2}}+{\frac {\sin {2x}}{4}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ac0a3ef626dd748a04c35bb801ab7f947a7451e)
![{\displaystyle \int \tan ^{n}x\,dx={\frac {1}{n-1}}\tan ^{n-1}x-\int \tan ^{n-2}x\,dx+C\quad \forall n\geq 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfca41aa46fca769d98b15795dd099b6b98a7629)
![{\displaystyle \int \tan ^{2}x\,dx=\tan x-x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7368fa8d3b731d365e3de153907e10e009ce754b)
![{\displaystyle \int \cot ^{n}x\,dx={\frac {1}{n-1}}\cot ^{n-1}x-\int \cot ^{n-2}x\,dx+C\quad \forall n\geq 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/edd8269a658252648e04c52783ea3c782465495a)
![{\displaystyle \int \cot ^{2}x\,dx=-\cot x-x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/019439035f1ac17982482fc94a1b5564872646b8)
![{\displaystyle \int \sec ^{n}x\,dx={\frac {1}{n-1}}\sec ^{n-2}x\tan x+{\frac {n-2}{n-1}}\int \sec ^{n-2}x\,dx+C\quad \forall n\geq 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a7fa20c3aa61a0c9591b762b6637da7490d0dca)
![{\displaystyle \int \csc ^{n}x\,dx=-{\frac {1}{n-1}}\csc ^{n-2}x\cot x+{\frac {n-2}{n-1}}\int \csc ^{n-2}x\,dx+C\quad \forall n\geq 2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd6fdadeaece9f227af304c3b8c34728d7e1841c)
逆三角関数を含む積分[編集]
![{\displaystyle \int \arcsin x\,dx=x\arcsin x+{\sqrt {1-x^{2}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ddfc7f7ceca71af5b218fa2e6e20ae52c521950)
![{\displaystyle \int \arccos x\,dx=x\arccos x-{\sqrt {1-x^{2}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed8f0fa234f076adac9b9aef92b59b4b4f3da43d)
![{\displaystyle \int \arctan x\,dx=x\arctan x-\ln {\sqrt {1+x^{2}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d6af0735a4e6bacece12f9f8821e0ede0b8dba9)
![{\displaystyle \int \operatorname {arccot} x\,dx=x\operatorname {arccot} x+\ln {\sqrt {1+x^{2}}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/255ff3af021daf9bbcfed364278301588c24020b)
![{\displaystyle \int \operatorname {arcsec} x\,dx=x\operatorname {arcsec} x-\ln(x-{\sqrt {x^{2}-1}})+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc82b83ceebd89c353c339ad59ad6258bc10410f)
![{\displaystyle \int \operatorname {arccsc} x\,dx=x\operatorname {arccsc} x+\ln(x+{\sqrt {x^{2}-1}})+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/987d31377abab16a9d3f258fce712133da8df17b)
指数関数を含む積分[編集]
![{\displaystyle \int e^{x}\,dx=e^{x}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e6d31e8ad38cc40b4e3d18ad17b756efa483abd)
![{\displaystyle \int \alpha ^{x}\,dx={\frac {\alpha ^{x}}{\ln \alpha }}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c89996ca547b099cd3ee63c14c8e03a303aef9a)
![{\displaystyle \int xe^{ax}\,dx={\frac {1}{a^{2}}}(ax-1)e^{ax}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cb6acd5c0f57563d2ed5f177868d319d3a342d6)
![{\displaystyle \int x^{n}e^{ax}\,dx={\frac {1}{a}}x^{n}e^{ax}-{\frac {n}{a}}\int x^{n-1}e^{ax}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9339ddc5a798ee6ad69b030b54e0b9b7208287c)
![{\displaystyle \int e^{ax}\sin bx\,dx={\frac {e^{ax}}{a^{2}+b^{2}}}(a\sin bx-b\cos bx)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/157e7fd733396dd95eca19f9d53842e756b2478f)
![{\displaystyle \int e^{ax}\cos bx\,dx={\frac {e^{ax}}{a^{2}+b^{2}}}(a\cos bx+b\sin bx)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab1381fe95fd3d8f3f507b2a686aaa4a05e25d79)
対数関数を含む積分[編集]
![{\displaystyle \int \ln x\,dx=x\ln x-x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54b730b8defb012d4140543f799c9fb1a7f1a3c8)
![{\displaystyle \int \log _{\alpha }x\,dx={\frac {1}{\ln \alpha }}\left({x\ln x-x}\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e67a4a3791595228e7d78ed96469e3eb0a897eff)
![{\displaystyle \int x^{n}\ln x\,dx={\frac {x^{n+1}}{(n+1)^{2}}}[(n+1)\ln x-1]+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/894d2f0d1cf257455556ecc979c5e2b725dab89e)
![{\displaystyle \int {\frac {1}{x\ln {x}}}\,dx=\ln {(\ln {x})}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5b2d16a6f290c9633ff819b56e8309b57bcf33d)
双曲線関数を含む積分[編集]
![{\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 \left(\cosh x\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b06396c494b3611b512933b246b5cff9d0b3b910)
![{\displaystyle \int \coth x\,dx=\ln \left(\sinh x\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b35167cd00e712a61de632594c6c99cf185ca3c3)
:グーデルマン関数
![{\displaystyle \int {\mbox{csch}}\ x\,dx=\ln \left(\tanh {x \over 2}\right)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ac3b99cf9955c7d2fee7d2712c0666a0aedaaac)
定積分[編集]
![{\displaystyle \int _{-\infty }^{\infty }e^{-\alpha x^{2}}\,dx={\sqrt {\frac {\pi }{\alpha }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3b3fbf55e064ab64f9d9fdd3057aee2337fe7d0)
![{\displaystyle \int _{0}^{\frac {\pi }{2}}{\mbox{sin}}^{n}x\,dx=\int _{0}^{\frac {\pi }{2}}{\mbox{cos}}^{n}x\,dx={\begin{cases}{\frac {n-1}{n}}\cdot {\frac {n-3}{n-2}}\cdot \cdots \cdot {\frac {4}{5}}\cdot {\frac {2}{3}},&{\mbox{if }}n>1{\mbox{ and }}n{\mbox{ is odd}}\\{\frac {n-1}{n}}\cdot {\frac {n-3}{n-2}}\cdot \cdots \cdot {\frac {3}{4}}\cdot {\frac {1}{2}}\cdot {\frac {\pi }{2}},&{\mbox{if }}n>0{\mbox{ and }}n{\mbox{ is even}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5adcde8295e45467da35bddb33a05c6658828416)
関連項目[編集]