Dit artikel bevat een lijst van integralen van irrationale functies. Integralen zijn het onderwerp van studie van de integraalrekening. De integralen in de lijst hieronder zijn veel voorkomende integralen van een functie onder de wortel. Er wordt van alle integralen de primitieve functie gegeven, maar de integratieconstante is in de uitkomst steeds weggelaten.
Integralen met
![{\displaystyle \int r\ \mathrm {d} x={\tfrac {1}{2}}(xr+a^{2}\ \ln(x+r))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49f815e7b6a96f6fbb6a5ff37c6d8a6e480dc350)
![{\displaystyle \int r^{3}\ \mathrm {d} x={\tfrac {1}{4}}xr^{3}+{\tfrac {3}{8}}a^{2}xr+{\tfrac {3}{8}}a^{4}\ln(x+r)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4aca32d58c75db48818bdda0637ff3d6e6a084f)
![{\displaystyle \int r^{5}\ \mathrm {d} x={\tfrac {1}{6}}xr^{5}+{\tfrac {5}{24}}a^{2}xr^{3}+{\tfrac {5}{16}}a^{4}xr+{\tfrac {5}{16}}a^{6}\ln(x+r)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94dacb7a7e175f32b925c446b8debc9b930d040a)
![{\displaystyle \int xr\ \mathrm {d} x={\tfrac {1}{3}}r^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9a192030dd0048ad434b61ed1ebebbbba1eb36)
![{\displaystyle \int xr^{3}\ \mathrm {d} x={\tfrac {1}{5}}r^{5}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37f235b3f83f5a09d45b5de26284e4b1b738071e)
![{\displaystyle \int xr^{2n+1}\ \mathrm {d} x={\frac {r^{2n+3}}{2n+3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f086ae8f96b6e71256212c41a917b7f96828d95)
![{\displaystyle \int x^{2}r\ \mathrm {d} x={\tfrac {1}{4}}xr^{3}-{\tfrac {1}{8}}a^{2}xr-{\tfrac {1}{8}}a^{4}\ln(x+r)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b68b0c0d4a5b393e0206ba13f532882029b4ed3c)
![{\displaystyle \int x^{2}r^{3}\ \mathrm {d} x={\tfrac {1}{6}}xr^{5}-{\tfrac {1}{24}}a^{2}xr^{3}-{\tfrac {1}{16}}a^{4}xr-{\tfrac {1}{16}}a^{6}\ln(x+r)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5beca78d8912f409e70c065ebfd09a74a9675bdb)
![{\displaystyle \int x^{3}r\ \mathrm {d} x={\tfrac {1}{5}}r^{5}-{\tfrac {1}{3}}a^{2}r^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/307a7058bbe9f491ec1c58d7889fdcebbe54ba56)
![{\displaystyle \int x^{3}r^{3}\ \mathrm {d} x={\tfrac {1}{7}}r^{7}-{\tfrac {1}{5}}a^{2}r^{5}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/752dc095d17c91601117f12dc59f30f83fbb6794)
![{\displaystyle \int x^{3}r^{2n+1}\ \mathrm {d} x={\frac {r^{2n+5}}{2n+5}}-{\frac {a^{3}r^{2n+3}}{2n+3}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60ed80b96af454998cf3f47d6b63a93b84cdd79d)
![{\displaystyle \int x^{4}r\ \mathrm {d} x={\tfrac {1}{6}}x^{3}r^{3}-{\tfrac {1}{8}}a^{2}xr^{3}+{\tfrac {1}{16}}a^{4}xr+{\tfrac {1}{16}}a^{6}\ln(x+r)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b38f5b267cd04dcdb4d2ba25acd7cd8bbfcd9fb8)
![{\displaystyle \int x^{4}r^{3}\ \mathrm {d} x={\tfrac {1}{8}}x^{3}r^{5}-{\tfrac {1}{16}}a^{2}xr^{5}+{\tfrac {1}{64}}a^{4}xr^{3}+{\tfrac {3}{128}}a^{6}xr+{\tfrac {3}{128}}a^{8}\ln(x+r)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2b9192f740b7b759e54924af95d188675273ffb)
![{\displaystyle \int x^{5}r\ \mathrm {d} x={\tfrac {1}{7}}r^{7}-{\tfrac {2}{5}}a^{2}r^{5}+{\tfrac {1}{3}}a^{4}r^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4322fb29471707957071592ee0d7d3617cca0440)
![{\displaystyle \int x^{5}r^{3}\ \mathrm {d} x={\tfrac {1}{9}}r^{9}-{\tfrac {2}{7}}a^{2}r^{7}+{\tfrac {1}{5}}a^{4}r^{5}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/724b0d02794ec41b3e37ad813f36148c97ffd2de)
![{\displaystyle \int x^{5}r^{2n+1}\ \mathrm {d} x={\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/4b8a01542b6ffaa7199419e53968f55050d544b9)
![{\displaystyle \int {\frac {r\ \mathrm {d} x}{x}}=r-a\ln \left|{\frac {a+r}{x}}\right|=r-a\ \operatorname {arsinh} {\frac {a}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a0d953c4fccf3d4edab023be433b987023d7f01)
![{\displaystyle \int {\frac {r^{3}\ \mathrm {d} x}{x}}={\tfrac {1}{3}}r^{3}+a^{2}r-a^{3}\ln \left|{\frac {a+r}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e056531b1b815457544278fc65dfce27a7549498)
![{\displaystyle \int {\frac {r^{5}\ \mathrm {d} x}{x}}={\tfrac {1}{5}}r^{5}+{\tfrac {1}{3}}a^{2}r^{3}+a^{4}r-a^{5}\ln \left|{\frac {a+r}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/730eecdc39a16629b6ced2dac562511c50f3f1c3)
![{\displaystyle \int {\frac {r^{7}\ \mathrm {d} x}{x}}={\tfrac {1}{7}}r^{7}+{\tfrac {1}{5}}a^{2}r^{5}+{\tfrac {1}{3}}a^{4}r^{3}+a^{6}r-a^{7}\ln \left|{\frac {a+r}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e71e1aa7ec00c9fff073d40e9b696f8c15804a1a)
![{\displaystyle \int {\frac {\mathrm {d} x}{r}}=\operatorname {arsinh} {\frac {x}{a}}=\ln \left({\frac {x+r}{a}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5298a366145831557bec85838f4097f1151fcc93)
![{\displaystyle \int {\frac {\mathrm {d} x}{r^{3}}}={\frac {x}{a^{2}r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c06d22f1453b2ce06b654f45be876bc7ffa3174)
![{\displaystyle \int {\frac {x\ \ \mathrm {d} x}{r}}=r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2402e316a451e5aeff995b996454766dde1d48)
![{\displaystyle \int {\frac {x\ \ \mathrm {d} x}{r^{3}}}=-{\frac {1}{r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b220689642c075e7b66fe11983d99edcba73750)
![{\displaystyle \int {\frac {x^{2}\ \mathrm {d} x}{r}}={\tfrac {1}{2}}xr-{\tfrac {1}{2}}a^{2}\ \operatorname {arsinh} {\frac {x}{a}}={\tfrac {1}{2}}xr-{\tfrac {1}{2}}a^{2}\ln \left({\frac {x+r}{a}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a7861277e752fea1015bce8905562f594903524)
![{\displaystyle \int {\frac {\mathrm {d} x}{xr}}=-{\frac {1}{a}}\ \operatorname {arsinh} {\frac {a}{x}}=-{\frac {1}{a}}\ln \left|{\frac {a+r}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1897b633b377f624627b385f1eda57d7803a3e05)
Integralen met
Voor de volgende integralen is
.
![{\displaystyle \int s\ \mathrm {d} x={\tfrac {1}{2}}(xs-a^{2}\ln(x+s))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0482866e51c8ed3b936409e9c33195a6005e1ab)
![{\displaystyle \int xs\ \mathrm {d} x={\tfrac {1}{3}}s^{3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92ef9ee15632fa3d17d244d924d2e9f0ba20f8dd)
![{\displaystyle \int {\frac {s\ \mathrm {d} x}{x}}=s-a\arccos \left|{\frac {a}{x}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b079a129da5b68acb0f0cf7deef92d6074ed943)
![{\displaystyle \int {\frac {\mathrm {d} x}{s}}=\int {\frac {\mathrm {d} x}{\sqrt {x^{2}-a^{2}}}}=\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5928f3627327f44b622b07e7239413a1726825ad)
Houd hier rekening met het feit dat
, waarbij alleen naar de positieve waarde moet worden gekeken, namelijk
![{\displaystyle \int {\frac {x\ \mathrm {d} x}{s}}=s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e89a1077f73ea595cfbad46de5885e11781e1aef)
![{\displaystyle \int {\frac {x\ \mathrm {d} x}{s^{3}}}=-{\frac {1}{s}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/663613fa5320b13307ad20f3cc05c3fe863ff0c3)
![{\displaystyle \int {\frac {x\ \mathrm {d} x}{s^{5}}}=-{\frac {1}{3s^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c24f5285181e219c32f17dd44df53ec42a0b7039)
![{\displaystyle \int {\frac {x\ \mathrm {d} x}{s^{7}}}=-{\frac {1}{5s^{5}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce400567c2375e81689a3ce41b4f6394193676ee)
![{\displaystyle \int {\frac {x\ \mathrm {d} x}{s^{2n+1}}}=-{\frac {1}{(2n-1)s^{2n-1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f20eb71aa9009b897d6d435882d8ac7ca480066)
![{\displaystyle \int {\frac {x^{2m}\ \mathrm {d} x}{s^{2n+1}}}=-{\frac {1}{2n-1}}{\frac {x^{2m-1}}{s^{2n-1}}}+{\frac {2m-1}{2n-1}}\int {\frac {x^{2m-2}\ \mathrm {d} x}{s^{2n-1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78024ac63e9739de1abbfa439793e82578136b01)
![{\displaystyle \int {\frac {x^{2}\ \mathrm {d} x}{s}}={\tfrac {1}{2}}xs+{\tfrac {1}{2}}a^{2}\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5206e52107442f16f192e8c62b3e3f335f39cfcc)
![{\displaystyle \int {\frac {x^{2}\ \mathrm {d} x}{s^{3}}}=-{\frac {x}{s}}+\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/310efefeeb4963eff89bd10fc27c29208e5805b5)
![{\displaystyle \int {\frac {x^{4}\ \mathrm {d} x}{s}}={\tfrac {1}{4}}x^{3}s+{\tfrac {3}{8}}a^{2}xs+{\tfrac {3}{8}}a^{4}\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c260f83e10e9cb7ddf72ad4512bae2cc8af8f11)
![{\displaystyle \int {\frac {x^{4}\ \mathrm {d} x}{s^{3}}}={\tfrac {1}{2}}xs-{\frac {a^{2}x}{s}}+{\tfrac {3}{2}}a^{2}\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4de90371c0688526acdf7f629d832efed26076e3)
![{\displaystyle \int {\frac {x^{4}\ \mathrm {d} x}{s^{5}}}=-{\frac {x}{s}}-{\frac {x^{3}}{3s^{3}}}+\ln \left|{\frac {x+s}{a}}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/670378c6ed9089c0edc052571aa951d454917811)
![{\displaystyle \int {\frac {x^{2m}\ \mathrm {d} x}{s^{2n+1}}}=(-1)^{n-m}{\frac {1}{a^{2(n-m)}}}\sum _{i=0}^{n-m-1}{\frac {1}{2(m+i)+1}}{n-m-1 \choose i}{\frac {x^{2(m+i)+1}}{s^{2(m+i)+1}}}\qquad n>m\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9293d312c34d19bd2bb6a0707990de565a14c030)
![{\displaystyle \int {\frac {\mathrm {d} x}{s^{3}}}=-{\frac {1}{a^{2}}}{\frac {x}{s}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62f4c28bc9eb94d2752f55ab373089c012d9cec4)
![{\displaystyle \int {\frac {\mathrm {d} x}{s^{5}}}={\frac {1}{a^{4}}}\left[{\frac {x}{s}}-{\tfrac {1}{3}}{\frac {x^{3}}{s^{3}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd705560c62f024beb029c160f94b55c41dc9422)
![{\displaystyle \int {\frac {\mathrm {d} x}{s^{7}}}=-{\frac {1}{a^{6}}}\left[{\frac {x}{s}}-{\tfrac {2}{3}}{\frac {x^{3}}{s^{3}}}+{\tfrac {1}{5}}{\frac {x^{5}}{s^{5}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a6f3edccad00497079e192ae699e960e152acb4)
![{\displaystyle \int {\frac {\mathrm {d} x}{s^{9}}}={\frac {1}{a^{8}}}\left[{\frac {x}{s}}-{\tfrac {3}{3}}{\frac {x^{3}}{s^{3}}}+{\tfrac {3}{5}}{\frac {x^{5}}{s^{5}}}-{\tfrac {1}{7}}{\frac {x^{7}}{s^{7}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e712fcf25118a8188eab07ee1cdbcf6fa4ae4659)
![{\displaystyle \int {\frac {x^{2}\ \mathrm {d} x}{s^{5}}}=-{\frac {1}{a^{2}}}{\frac {x^{3}}{3s^{3}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1841fd176d3f3f453f032e0d291d92a530abec4)
![{\displaystyle \int {\frac {x^{2}\ \mathrm {d} x}{s^{7}}}={\frac {1}{a^{4}}}\left[{\tfrac {1}{3}}{\frac {x^{3}}{3s^{3}}}-{\tfrac {1}{5}}{\frac {x^{5}}{s^{5}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8144c9e828d29a8e0537fd6d032a515d6ad8a19a)
![{\displaystyle \int {\frac {x^{2}\ \mathrm {d} x}{s^{9}}}=-{\frac {1}{a^{6}}}\left[{\tfrac {1}{3}}{\frac {x^{3}}{s^{3}}}-{\tfrac {2}{5}}{\frac {x^{5}}{s^{5}}}+{\tfrac {1}{7}}{\frac {x^{7}}{s^{7}}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7bd7215c6045dc7365d33c5c6a50e72d33b8647b)
Integralen waarbij
![{\displaystyle \int u\ \mathrm {d} x={\frac {1}{2}}\left(xu+a^{2}\arcsin {\frac {x}{a}}\right)\qquad |x|\leq |a|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9206208a890802b7998e24356714993605b288e1)
![{\displaystyle \int xu\ \mathrm {d} x=-{\frac {1}{3}}u^{3}\qquad (|x|\leq |a|)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52dba848dad6333f426d53397e2faf5daa3e8139)
![{\displaystyle \int x^{2}u\ \mathrm {d} x=-{\frac {x}{4}}u^{3}+{\frac {a^{2}}{8}}(xu+a^{2}\arcsin {\frac {x}{a}})\qquad |x|\leq |a|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44299d7a5b06d4c76ee87e72804be9f865e50a13)
![{\displaystyle \int {\frac {u\ \mathrm {d} x}{x}}=u-a\ln \left|{\frac {a+u}{x}}\right|\qquad |x|\leq |a|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/547553c1e31e93a72598a8cfdf739ca466b372eb)
![{\displaystyle \int {\frac {\mathrm {d} x}{u}}=\arcsin {\frac {x}{a}}\qquad (|x|\leq |a|)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24a7a8d59c2b6152d49a4d23b35b32e172dbe7af)
![{\displaystyle \int {\frac {x^{2}\ \mathrm {d} x}{u}}={\frac {1}{2}}\left(-xu+a^{2}\arcsin {\frac {x}{a}}\right)\qquad |x|\leq |a|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cbaf8069741e6b4992ba2dd386dda5e0e2aeb71)
![{\displaystyle \int u\ \mathrm {d} x={\frac {1}{2}}\left(xu-\operatorname {sgn} x\ \operatorname {arcosh} \left|{\frac {x}{a}}\right|\right)\qquad {\mbox{voor }}|x|\geq |a|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fed3e12e52a676852b6b5b952b7e4d2c39f778a7)
![{\displaystyle \int {\frac {x}{u}}\ \mathrm {d} x=-u\qquad |x|\leq |a|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd768453d0495129f86abd68eaf1e2adbafb14a2)
Integralen waarbij
Neem aan dat
niet kan worden geschreven als de verkorte vorm
![{\displaystyle \int {\frac {\mathrm {d} x}{R}}={\frac {1}{\sqrt {a}}}\ln \left|2{\sqrt {a}}R+2ax+b\right|\qquad {\mbox{voor }}a>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/926cc6079f7e71edf9369f9df2b10c3e3bd31b5c)
![{\displaystyle \int {\frac {\mathrm {d} x}{R}}={\frac {1}{\sqrt {a}}}\ \operatorname {arsinh} {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{voor }}a>0,\ 4ac-b^{2}>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0bfbb398dbbbfc8d15bf2c89b7077bff423b7e8)
![{\displaystyle \int {\frac {\mathrm {d} x}{R}}={\frac {1}{\sqrt {a}}}\ln |2ax+b|\quad {\mbox{voor }}a>0,\ 4ac-b^{2}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2e5a0413c70d3717bdabdd9a9e266e812cc233a)
![{\displaystyle \int {\frac {\mathrm {d} x}{R}}=-{\frac {1}{\sqrt {-a}}}\arcsin {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad {\mbox{voor }}a<0,\ 4ac-b^{2}<0,\ |2ax+b|<{\sqrt {b^{2}-4ac}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49d92445121c38cc0d17374375da87fdb4492aaf)
![{\displaystyle \int {\frac {\mathrm {d} x}{R^{3}}}={\frac {4ax+2b}{(4ac-b^{2})R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/985a536a45f635b638a2751b48a63d03210904fd)
![{\displaystyle \int {\frac {\mathrm {d} x}{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/4f99db40a2c4c701e03e92ce0299ffb38282a7cc)
![{\displaystyle \int {\frac {\mathrm {d} x}{R^{2n+1}}}={\frac {2}{(2n-1)(4ac-b^{2})}}\left({\frac {2ax+b}{R^{2n-1}}}+4a(n-1)\int {\frac {\mathrm {d} x}{R^{2n-1}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/871e9e1df0c62b671281de72806c0c4db3d91b18)
![{\displaystyle \int {\frac {x}{R}}\ \mathrm {d} x={\frac {R}{a}}-{\frac {b}{2a}}\int {\frac {\mathrm {d} x}{R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92b90e2546ad246f4f4247b42802239908365f9a)
![{\displaystyle \int {\frac {x}{R^{3}}}\ \mathrm {d} x=-{\frac {2bx+4c}{(4ac-b^{2})R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0cfff81a5a3c756053ab59e599f1b09f8ce31cb4)
![{\displaystyle \int {\frac {x}{R^{2n+1}}}\ \mathrm {d} x=-{\frac {1}{(2n-1)aR^{2n-1}}}-{\frac {b}{2a}}\int {\frac {\mathrm {d} x}{R^{2n+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e4a43287e323cd890a4321b4b9a134630ca8bbd)
![{\displaystyle \int {\frac {\mathrm {d} x}{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/e0f20cfd01d990937deca849ea3746c12ace94a0)
![{\displaystyle \int {\frac {\mathrm {d} x}{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/bffbb061f5bf17c2d1882e5ad51f1d17f2513f8d)
Integralen waarbij
![{\displaystyle \int S{\mathrm {d} x}={\frac {2S^{3}}{3a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95332a45f4ca4035ba2dac1dcaa7c7642880131a)
![{\displaystyle \int {\frac {\mathrm {d} x}{S}}={\frac {2S}{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba5277866b1aeb9984b3ad1a12adbf42bd571e13)
![{\displaystyle \int {\frac {\mathrm {d} x}{xS}}={\begin{cases}{\mbox{voor }}b>0,\quad ax>0\\-{\frac {2}{\sqrt {b}}}\mathrm {artanh} \left({\frac {S}{\sqrt {b}}}\right)&{\mbox{voor }}b>0,\quad ax<0\\{\frac {2}{\sqrt {-b}}}\arctan \left({\frac {S}{\sqrt {-b}}}\right)&{\mbox{voor }}b<0\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a50351922b8672b67427eaee1358a688aef8f899)
![{\displaystyle \int {\frac {S}{x}}\ \mathrm {d} x={\begin{cases}2\left(S-{\sqrt {b}}\ \mathrm {arcoth} \left({\frac {S}{\sqrt {b}}}\right)\right)&{\mbox{voor }}b>0,\quad ax>0\\2\left(S-{\sqrt {b}}\ \mathrm {artanh} \left({\frac {S}{\sqrt {b}}}\right)\right)&{\mbox{voor }}b>0,\quad ax<0\\2\left(S-{\sqrt {-b}}\arctan \left({\frac {S}{\sqrt {-b}}}\right)\right)&{\mbox{voor }}b<0\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9216c51533822ec0200b7a8c2d4201441ea40003)
![{\displaystyle \int {\frac {x^{n}}{S}}\ \mathrm {d} x={\frac {2}{a(2n+1)}}\left(x^{n}S-bn\int {\frac {x^{n-1}}{S}}\ \mathrm {d} x\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db5f1c6d852a4dd73867ccd30c9b244ab06a0c7e)
![{\displaystyle \int x^{n}S\ \mathrm {d} x={\frac {2}{a(2n+3)}}\left(x^{n}S^{3}-nb\int x^{n-1}S\ \mathrm {d} x\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6aeba50baea7fc18ca012bbb494cd245617a1b4)
![{\displaystyle \int {\frac {1}{x^{n}S}}\ \mathrm {d} x=-{\frac {1}{b(n-1)}}\left({\frac {S}{x^{n-1}}}+\left(n-{\frac {3}{2}}\right)a\int {\frac {\mathrm {d} x}{x^{n-1}S}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58b32f725a75b7197bfd5800011021f0dc72476b)
Literatuur
- S Gradshteyn, IM Ryzhik, A Jeffrey en D Zwillinger. Table of Integrals, Series, and Products, 2007. met te downloaden pdf's, ISBN 978-0-12-373637-6.