List of integrals of irrational functions
The following is a list of integrals (antiderivative functions) of irrational functions. For a complete list of integral functions, see lists of integrals. Throughout this article the constant of integration is omitted for brevity.
Integrals involving r = √a2 + x2
![{\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/757ff9ec03548bc267baa1e9b62ae09ba29959cb)
![{\displaystyle \int r^{3}\,dx={\frac {1}{4}xr^{3}+{\frac {3}{8}a^{2}xr+{\frac {3}{8}a^{4}\ln \left(x+r\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7aa63739c00a1763e88c3e3e11a95623d1dc2d7f)
![{\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(x+r\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3616d6c3889707e4405b23e6b38031e57d91ed8)
![{\displaystyle \int xr\,dx={\frac {r^{3}{3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebb71452dba540cbbb7e94426b63fb5268c47c7f)
![{\displaystyle \int xr^{3}\,dx={\frac {r^{5}{5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77019dd2afd82923a973aa2f7680a97e3df0aea8)
![{\displaystyle \int xr^{2n+1}\,dx={\frac {r^{2n+3}{2n+3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fa545554b384f332521236696efc27f8b50d977)
![{\displaystyle \int x^{2}r\,dx={\frac {xr^{3}{4}-{\frac {a^{2}xr}{8}-{\frac {a^{4}{8}\ln \left(x+r\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59c9349d229ac808d13a97e07fb51a9508b6e5e3)
![{\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(x+r\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bd383ab4eb3dee757db6fac447aa5522605bef0)
![{\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/8886c9df29f7cb8a9fbe5dc694b74b3f1590dafb)
![{\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/f5cceb1a3f698a38b012069e9e8d6be32ef520f1)
![{\displaystyle \int x^{3}r^{2n+1}\,dx={\frac {r^{2n+5}{2n+5}-{\frac {a^{2}r^{2n+3}{2n+3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c37578aa1309d6ae1b072a50eb9dea0251a0598a)
![{\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(x+r\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf79a23b07e2b620e77a921344127408551c0fbe)
![{\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(x+r\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffc497032d7cb34e5ab01a227b5747cbf4910a76)
![{\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/d154741fca5708cbd7d062fed335a69b8a5c7c5e)
![{\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/f4cb148de4e334085cdbe18ce4f45659925082f8)
![{\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/f5a5ddb75d41b54c1ab174333d7c6aea321f94d8)
![{\displaystyle \int {\frac {r\,dx}{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/5a511b55b2624c38325653d4f6e9b292fae2da49)
![{\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/4508f8d7ecb7336d153425ded40ace67988347c9)
![{\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/05ca1d896950595916eb13991c5da85427145217)
![{\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/e7f07e4f96d167d6d54ebf56a9209833bc4bb5a3)
![{\displaystyle \int {\frac {dx}{r}=\operatorname {arsinh} {\frac {x}{a}=\ln \left({\frac {x+r}{a}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54a78aad92f1fd09df4f4c33d5b28081aec09c45)
![{\displaystyle \int {\frac {dx}{r^{3}={\frac {x}{a^{2}r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e903aacbfd829184b3eeb9233ce14ac236d1e6b)
![{\displaystyle \int {\frac {x\,dx}{r}=r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77a3156a33834a19e192af67f3a54da6ddbe4ce1)
![{\displaystyle \int {\frac {x\,dx}{r^{3}=-{\frac {1}{r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8efc8a15f3072555233b2781dc4c398780f904e6)
![{\displaystyle \int {\frac {x^{2}\,dx}{r}={\frac {x}{2}r-{\frac {a^{2}{2}\,\operatorname {arsinh} {\frac {x}{a}={\frac {x}{2}r-{\frac {a^{2}{2}\ln \left({\frac {x+r}{a}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f811e33067f8d779d7ae5dfa3a842e7e69a1b366)
![{\displaystyle \int {\frac {dx}{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/7217f767426c8ffb2d25674cf08d8d9c1a30035e)
Integrals involving s = √x2 − a2
Assume x2 > a2 (for x2 < a2, see next section):
![{\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/35886442d8a417195c070de09e947bd11046f6a7)
![{\displaystyle \int xs\,dx={\frac {1}{3}s^{3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79ec50724959547d7a180296ed822323ac0dea9a)
![{\displaystyle \int {\frac {s\,dx}{x}=s-|a|\arccos \left|{\frac {a}{x}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61795cdfbe6ed4953270eacbc923f4c60bda6db5)
Here
where the positive value of
is to be taken.
![{\displaystyle \int {\frac {dx}{xs}={\frac {1}{a}\operatorname {arcsec} \left|{\frac {x}{a}\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e01fd7bb771dceca4f0d713cf13fbb7e5db1eebf)
![{\displaystyle \int {\frac {x\,dx}{s}=s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7498e24f5b810335c07522c0a9739358efc171ce)
![{\displaystyle \int {\frac {x\,dx}{s^{3}=-{\frac {1}{s}](https://wikimedia.org/api/rest_v1/media/math/render/svg/420ee527cc619eaea1f897b2d8d9765e2ac3a34a)
![{\displaystyle \int {\frac {x\,dx}{s^{5}=-{\frac {1}{3s^{3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da2899f813091ab868b134dbb398651fd67cdd4a)
![{\displaystyle \int {\frac {x\,dx}{s^{7}=-{\frac {1}{5s^{5}](https://wikimedia.org/api/rest_v1/media/math/render/svg/534b1e0f5248dfd33193173d280a6ab0afa73d30)
![{\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/df23be08dda0b35eeec793c80b44ef5db030dc30)
![{\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/e146edfd566e4448d1d12d5bd6abeff171019619)
![{\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/e3984e353316fd0845d7e2a525a3c20cdf3b3fef)
![{\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/19429e0acc1777fdde820e9fbf863900c3c3eb17)
![{\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/f5139c41de6581d88b5271f6f4813caafaf4dd1a)
![{\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/f54a7b455636a1059cf15874b29fb564e528b5de)
![{\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/0cc599315c9d3c1c1d2a0cf582c1e89c3efc93b3)
![{\displaystyle \int {\frac {x^{2m}\,dx}{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 {\mbox{(}n>m\geq 0{\mbox{)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1b4e7668e09862fe1b92881767247b577cf2d85)
![{\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/f7678ddbcc74493dcef267531c411de750e6ecfc)
![{\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/6a98057cf3f3d6b7025114445c972bb6b7b7af9d)
![{\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/9cce4b87e7a47ce42042803038139f830afd5d37)
Integrals involving u = √a2 − x2
![{\displaystyle \int u\,dx={\frac {1}{2}\left(xu+a^{2}\arcsin {\frac {x}{a}\right)\qquad {\mbox{(}|x|\leq |a|{\mbox{)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbf908b7dc2ed6a0e0e1763ecd5418b2f374b1b8)
![{\displaystyle \int xu\,dx=-{\frac {1}{3}u^{3}\qquad {\mbox{(}|x|\leq |a|{\mbox{)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59db13d03287959afa947e1496b77be923eb8803)
![{\displaystyle \int x^{2}u\,dx=-{\frac {x}{4}u^{3}+{\frac {a^{2}{8}(xu+a^{2}\arcsin {\frac {x}{a})\qquad {\mbox{(}|x|\leq |a|{\mbox{)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5768e21637b0760808486ab008ce28e37ddcb1a)
![{\displaystyle \int {\frac {u\,dx}{x}=u-a\ln \left|{\frac {a+u}{x}\right|\qquad {\mbox{(}|x|\leq |a|{\mbox{)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08989a6d3cda85094876242fabe63eeecc9704b4)
![{\displaystyle \int {\frac {dx}{u}=\arcsin {\frac {x}{a}\qquad {\mbox{(}|x|\leq |a|{\mbox{)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dacefb8fb4df4a4cd90b7e24706b39ce53a66023)
![{\displaystyle \int {\frac {x^{2}\,dx}{u}={\frac {1}{2}\left(-xu+a^{2}\arcsin {\frac {x}{a}\right)\qquad {\mbox{(}|x|\leq |a|{\mbox{)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/989d8563dd9195878ff02d72ac82861a9c48e286)
![{\displaystyle \int u\,dx={\frac {1}{2}\left(xu-\operatorname {sgn} x\,\operatorname {arcosh} \left|{\frac {x}{a}\right|\right)\qquad {\mbox{(for }|x|\geq |a|{\mbox{)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2845ce5de5f507e697fd6b0843d2000ada598b24)
![{\displaystyle \int {\frac {x}{u}\,dx=-u\qquad {\mbox{(}|x|\leq |a|{\mbox{)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/641d5d85a29525355dfcf37fe4546054300cc04d)
Integrals involving R = √ax2 + bx + c
Assume (ax2 + bx + c) cannot be reduced to the following expression (px + q)2 for some p and q.
![{\displaystyle \int {\frac {dx}{R}={\frac {1}{\sqrt {a}\ln \left|2{\sqrt {a}R+2ax+b\right|\qquad {\mbox{(for }a>0{\mbox{)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3add2ea6d0465dc1f8f3de7193104f8bad5b7a4b)
![{\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/5b4017a60a8edb505fe2149a772d5e231c1f1ed9)
![{\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/e6978eb76d4e7435f227a36d75284fd5ac366a76)
![{\displaystyle \int {\frac {x}{R^{3}\,dx=-{\frac {2bx+4c}{(4ac-b^{2})R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9364096dd1c0621646aefef2705945240a52650)
![{\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/1160bb82c90871b7587728777ee03f9c59953757)
![{\displaystyle \int {\frac {dx}{xR}=-{\frac {1}{\sqrt {c}\ln \left|{\frac {2{\sqrt {c}R+bx+2c}{x}\right|,~c>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e62f8d2d2edda5f638ea40d69da7d9b4ee20dbfe)
![{\displaystyle \int {\frac {dx}{xR}=-{\frac {1}{\sqrt {c}\operatorname {arsinh} \left({\frac {bx+2c}{|x|{\sqrt {4ac-b^{2}\right),~c<0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94e44a3b8a2baaecefe1426abe9f66b483bae82d)
![{\displaystyle \int {\frac {dx}{xR}={\frac {1}{\sqrt {-c}\operatorname {arcsin} \left({\frac {bx+2c}{|x|{\sqrt {b^{2}-4ac}\right),~c<0,b^{2}-4ac>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2140db710a6f09b4bc6cd2f81d79b7c44d325790)
![{\displaystyle \int {\frac {dx}{xR}=-{\frac {2}{bx}\left({\sqrt {ax^{2}+bx}\right),~c=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0121ed272243c4b2169efdf39fcbf0ea6fcf1ed)
![{\displaystyle \int {\frac {x^{2}{R}\,dx={\frac {2ax-3b}{4a^{2}R+{\frac {3b^{2}-4ac}{8a^{2}\int {\frac {dx}{R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fed9e5a7ab71563eb0fbaadcb9d0c92c0078eaf)
![{\displaystyle \int {\frac {dx}{x^{2}R}=-{\frac {R}{cx}-{\frac {b}{2c}\int {\frac {dx}{xR}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c7ed093e910b3e384b46e617eb19d4895834e2f3)
![{\displaystyle \int R\,dx={\frac {2ax+b}{4a}R+{\frac {4ac-b^{2}{8a}\int {\frac {dx}{R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ca86e8e08abf7030acd7d02e1fd1cf114242120)
![{\displaystyle \int xR\,dx={\frac {R^{3}{3a}-{\frac {b(2ax+b)}{8a^{2}R-{\frac {b(4ac-b^{2})}{16a^{2}\int {\frac {dx}{R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cddad583bf46e028aa94c79b5ea041fb37319ad)
![{\displaystyle \int x^{2}R\,dx={\frac {6ax-5b}{24a^{2}R^{3}+{\frac {5b^{2}-4ac}{16a^{2}\int R\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98a7ef1d38a79135f43c0bec8b994d9a9f6b6ec3)
![{\displaystyle \int {\frac {R}{x}\,dx=R+{\frac {b}{2}\int {\frac {dx}{R}+c\int {\frac {dx}{xR}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24f35cd51c18e0a4c87424a2b6c30332a492d7e7)
![{\displaystyle \int {\frac {R}{x^{2}\,dx=-{\frac {R}{x}+a\int {\frac {dx}{R}+{\frac {b}{2}\int {\frac {dx}{xR}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70b2c31c5124e627a2f7e19a1ed23cd412b85bd6)
![{\displaystyle \int {\frac {x^{2}\,dx}{R^{3}={\frac {(2b^{2}-4ac)x+2bc}{a(4ac-b^{2})R}+{\frac {1}{a}\int {\frac {dx}{R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6bd1d6dd570c6d9365debd2938e29a1a6646f696)
Integrals involving S = √ax + b
![{\displaystyle \int S\,dx={\frac {2S^{3}{3a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cee123de6f144edd243ba2b239d3986700183d72)
![{\displaystyle \int {\frac {dx}{S}={\frac {2S}{a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6458c8a42c9321a984f3efc64b9045abd706ca70)
![{\displaystyle \int {\frac {dx}{xS}={\begin{cases}-{\dfrac {2}{\sqrt {b}\operatorname {arcoth} \left({\dfrac {S}{\sqrt {b}\right)&{\mbox{(for }b>0,\quad ax>0{\mbox{)}\\-{\dfrac {2}{\sqrt {b}\operatorname {artanh} \left({\dfrac {S}{\sqrt {b}\right)&{\mbox{(for }b>0,\quad ax<0{\mbox{)}\\{\dfrac {2}{\sqrt {-b}\arctan \left({\dfrac {S}{\sqrt {-b}\right)&{\mbox{(for }b<0{\mbox{)}\\\end{cases}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1449f96b3c8cfa7b2ae075541c0eb076e122c5f6)
![{\displaystyle \int {\frac {S}{x}\,dx={\begin{cases}2\left(S-{\sqrt {b}\,\operatorname {arcoth} \left({\dfrac {S}{\sqrt {b}\right)\right)&{\mbox{(for }b>0,\quad ax>0{\mbox{)}\\2\left(S-{\sqrt {b}\,\operatorname {artanh} \left({\dfrac {S}{\sqrt {b}\right)\right)&{\mbox{(for }b>0,\quad ax<0{\mbox{)}\\2\left(S-{\sqrt {-b}\arctan \left({\dfrac {S}{\sqrt {-b}\right)\right)&{\mbox{(for }b<0{\mbox{)}\\\end{cases}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e890c728ff6f9cb934acc74bb884650c4dbb98be)
![{\displaystyle \int {\frac {x^{n}{S}\,dx={\frac {2}{a(2n+1)}\left(x^{n}S-bn\int {\frac {x^{n-1}{S}\,dx\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/220fa89474d8cd3fb5b29eaae008b5e630411a40)
![{\displaystyle \int x^{n}S\,dx={\frac {2}{a(2n+3)}\left(x^{n}S^{3}-nb\int x^{n-1}S\,dx\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e352ae95c63969fb203a29980f43708691cdae29)
![{\displaystyle \int {\frac {1}{x^{n}S}\,dx=-{\frac {1}{b(n-1)}\left({\frac {S}{x^{n-1}+\left(n-{\frac {3}{2}\right)a\int {\frac {dx}{x^{n-1}S}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc2bfa895a4d84e126d997c1f76d09f0712a321a)
References
- Abramowitz, Milton; Stegun, Irene A., eds. (1972). "Chapter 3". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover.
- Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. ISBN 978-0-12-384933-5. LCCN 2014010276. (Several previous editions as well.)
- Peirce, Benjamin Osgood (1929) [1899]. "Chapter 3". A Short Table of Integrals (3rd revised ed.). Boston: Ginn and Co. pp. 16–30.