La siguiente es una lista de integrales ( funciones antiderivadas ) de funciones irracionales . Para obtener una lista completa de funciones integrales, consulte listas de integrales . A lo largo de este artículo se omite la constante de integración por motivos de brevedad.
Integrales que involucran r = √ a 2 + x 2
![{\displaystyle \int r\,dx={\frac {1}{2}}\left(xr+a^{2}\,\ln \left(x+r\right)\right)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int xr\,dx={\frac {r^{3}}{3}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int xr^{3}\,dx={\frac {r^{5}}{5}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int xr^{2n+1}\,dx={\frac {r^{2n+3}}{2n+3}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int x^{2}r\,dx={\frac {xr^{3}}{4}}-{\frac {a^{2}xr}{8}}-{\frac { a^{4}}{8}}\ln \left(x+r\right)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int x^{3}r\,dx={\frac {r^{5}}{5}}-{\frac {a^{2}r^{3}}{3}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int x^{3}r^{3}\,dx={\frac {r^{7}}{7}}-{\frac {a^{2}r^{5}}{ 5}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int x^{3}r^{2n+1}\,dx={\frac {r^{2n+5}}{2n+5}}-{\frac {a^{2}r ^{2n+3}}{2n+3}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int x^{5}r\,dx={\frac {r^{7}}{7}}-{\frac {2a^{2}r^{5}}{5}}+ {\frac {a^{4}r^{3}}{3}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int x^{5}r^{3}\,dx={\frac {r^{9}}{9}}-{\frac {2a^{2}r^{7}}{ 7}}+{\frac {a^{4}r^{5}}{5}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {r\,dx}{x}}=ra\ln \left|{\frac {a+r}{x}}\right|=ra\,\operatorname {arsinh} { \frac{a}{x}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {r^{3}\,dx}{x}}={\frac {r^{3}}{3}}+a^{2}ra^{3}\ln \izquierda|{\frac {a+r}{x}}\derecha|}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {r^{5}\,dx}{x}}={\frac {r^{5}}{5}}+{\frac {a^{2}r^{ 3}}{3}}+a^{4}ra^{5}\ln \left|{\frac {a+r}{x}}\right|}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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}ra^{7}\ln \left|{\frac {a+r }{x}}\derecha|}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {dx}{r}}=\operatorname {arsinh} {\frac {x}{a}}=\ln \left({\frac {x+r}{a}}\ bien)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {dx}{r^{3}}}={\frac {x}{a^{2}r}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {x\,dx}{r}}=r}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {x\,dx}{r^{3}}}=-{\frac {1}{r}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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}}\derecha)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {dx}{xr}}=-{\frac {1}{a}}\,\operatorname {arsinh} {\frac {a}{x}}=-{\frac { 1}{a}}\ln \left|{\frac {a+r}{x}}\right|}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Integrales que involucran s = √ x 2 − a 2
Supongamos que x 2 > a 2 (para x 2 < a 2 , consulte la siguiente sección):
![{\displaystyle \int s\,dx={\frac {1}{2}}\left(xs-a^{2}\ln \left|x+s\right|\right)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int xs\,dx={\frac {1}{3}}s^{3}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {s\,dx}{x}}=s-|a|\arccos \left|{\frac {a}{x}}\right|}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Aquí donde se debe tomar el valor positivo de .![{\displaystyle \ln \left|{\frac {x+s}{a}}\right|=\operatorname {sgn} (x)\,\operatorname {arcosh} \left|{\frac {x}{a }}\right|={\frac {1}{2}}\ln \left({\frac {x+s}{xs}}\right)\,,}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \operatorname {arcosh} \left|{\frac {x}{a}}\right|}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {dx}{xs}}={\frac {1}{a}}\operatorname {arcsec} \left|{\frac {x}{a}}\right|}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {x\,dx}{s}}=s}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {x\,dx}{s^{3}}}=-{\frac {1}{s}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {x\,dx}{s^{5}}}=-{\frac {1}{3s^{3}}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {x\,dx}{s^{7}}}=-{\frac {1}{5s^{5}}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {x\,dx}{s^{2n+1}}}=-{\frac {1}{(2n-1)s^{2n-1}}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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}}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {x^{2}\,dx}{s}}={\frac {xs}{2}}+{\frac {a^{2}}{2}}\ln \izquierda|{\frac {x+s}{a}}\derecha|}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {x^{2}\,dx}{s^{3}}}=-{\frac {x}{s}}+\ln \left|{\frac {x+ s}{a}}\derecha|}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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|}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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|}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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|}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {x^{2m}\,dx}{s^{2n+1}}}=(-1)^{nm}{\frac {1}{a^{2(nm) )}}}\sum _{i=0}^{nm-1}{\frac {1}{2(m+i)+1}}{nm-1 \choose i}{\frac {x^{ 2(m+i)+1}}{s^{2(m+i)+1}}}\qquad {\mbox{(}}n>m\geq 0{\mbox{)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {dx}{s^{3}}}=-{\frac {1}{a^{2}}}{\frac {x}{s}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {dx}{s^{5}}}={\frac {1}{a^{4}}}\left[{\frac {x}{s}}-{\ frac {1}{3}}{\frac {x^{3}}{s^{3}}}\right]}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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}}}\derecha]}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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]}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {x^{2}\,dx}{s^{5}}}=-{\frac {1}{a^{2}}}{\frac {x^{3 }}{3s^{3}}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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}} }\bien]}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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]}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Integrales que involucran u = √ a 2 − x 2
![{\displaystyle \int u\,dx={\frac {1}{2}}\left(xu+a^{2}\arcsin {\frac {x}{a}}\right)\qquad {\mbox {(}}|x|\leq |a|{\mbox{)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int xu\,dx=-{\frac {1}{3}}u^{3}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}} }](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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{)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {u\,dx}{x}}=ua\ln \left|{\frac {a+u}{x}}\right|\qquad {\mbox{(}}| x|\leq |a|{\mbox{)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {dx}{u}}=\arcsin {\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{) }}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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{)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int u\,dx={\frac {1}{2}}\left(xu-\operatorname {sgn} x\,\operatorname {arcosh} \left|{\frac {x}{a} }\right|\right)\qquad {\mbox{(para }}|x|\geq |a|{\mbox{)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {x}{u}}\,dx=-u\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Integrales que involucran R = √ ax 2 + bx + c
Supongamos que ( ax 2 + bx + c ) no se puede reducir a la siguiente expresión ( px + q ) 2 para algunos p y q .
![{\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\ln \left|2{\sqrt {a}}R+2ax+b\right |\qquad {\mbox{(para }}a>0{\mbox{)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\,\operatorname {arsinh} {\frac {2ax+b}{\sqrt {4ac -b^{2}}}}\qquad {\mbox{(para }}a>0{\mbox{, }}4ac-b^{2}>0{\mbox{)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\ln |2ax+b|\quad {\mbox{(para }}a>0 {\mbox{, }}4ac-b^{2}=0{\mbox{)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {dx}{R}}=-{\frac {1}{\sqrt {-a}}}\arcsin {\frac {2ax+b}{\sqrt {b^{2 }-4ac}}}\qquad {\mbox{(para }}a<0{\mbox{, }}4ac-b^{2}<0{\mbox{, }}\left|2ax+b\right |<{\sqrt {b^{2}-4ac}}{\mbox{)}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {dx}{R^{3}}}={\frac {4ax+2b}{(4ac-b^{2})R}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {x}{R}}\,dx={\frac {R}{a}}-{\frac {b}{2a}}\int {\frac {dx}{R }}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {x}{R^{3}}}\,dx=-{\frac {2bx+4c}{(4ac-b^{2})R}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {x}{R^{2n+1}}}\,dx=-{\frac {1}{(2n-1)aR^{2n-1}}}-{\ frac {b}{2a}}\int {\frac {dx}{R^{2n+1}}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {dx}{xR}}=-{\frac {1}{\sqrt {c}}}\ln \left|{\frac {2{\sqrt {c}}R+ bx+2c}{x}}\derecha|,~c>0}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {dx}{xR}}=-{\frac {1}{\sqrt {c}}}\operatorname {arsinh} \left({\frac {bx+2c}{|x |{\sqrt {4ac-b^{2}}}}}\right),~c<0}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {dx}{xR}}=-{\frac {2}{bx}}\left({\sqrt {ax^{2}+bx}}\right),~c= 0}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {x^{2}}{R}}\,dx={\frac {2ax-3b}{4a^{2}}}R+{\frac {3b^{2}- 4ac}{8a^{2}}}\int {\frac {dx}{R}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {dx}{x^{2}R}}=-{\frac {R}{cx}}-{\frac {b}{2c}}\int {\frac {dx {xR}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int R\,dx={\frac {2ax+b}{4a}}R+{\frac {4ac-b^{2}}{8a}}\int {\frac {dx}{R} }}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int x^{2}R\,dx={\frac {6ax-5b}{24a^{2}}}R^{3}+{\frac {5b^{2}-4ac}{ 16a^{2}}}\int R\,dx}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {R}{x}}\,dx=R+{\frac {b}{2}}\int {\frac {dx}{R}}+c\int {\frac { dx}{xR}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {R}{x^{2}}}\,dx=-{\frac {R}{x}}+a\int {\frac {dx}{R}}+{ \frac {b}{2}}\int {\frac {dx}{xR}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Integrales que involucran S = √ ax + b
![{\displaystyle \int S\,dx={\frac {2S^{3}}{3a}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {dx}{S}}={\frac {2S}{a}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {dx}{xS}}={\begin{casos}-{\dfrac {2}{\sqrt {b}}}\operatorname {arcoth} \left({\dfrac {S }{\sqrt {b}}}\right)&{\mbox{(para }}b>0,\quad ax>0{\mbox{)}}\\-{\dfrac {2}{\sqrt { b}}}\operatorname {artanh} \left({\dfrac {S}{\sqrt {b}}}\right)&{\mbox{(para }}b>0,\quad ax<0{\mbox {)}}\\{\dfrac {2}{\sqrt {-b}}}\arctan \left({\dfrac {S}{\sqrt {-b}}}\right)&{\mbox{( para }}b<0{\mbox{)}}\\\end{casos}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int {\frac {S}{x}}\,dx={\begin{cases}2\left(S-{\sqrt {b}}\,\operatorname {arcoth} \left({\ dfrac {S}{\sqrt {b}}}\right)\right)&{\mbox{(para }}b>0,\quad ax>0{\mbox{)}}\\2\left(S -{\sqrt {b}}\,\operatorname {artanh} \left({\dfrac {S}{\sqrt {b}}}\right)\right)&{\mbox{(para }}b>0 ,\quad ax<0{\mbox{)}}\\2\left(S-{\sqrt {-b}}\arctan \left({\dfrac {S}{\sqrt {-b}}}\ right)\right)&{\mbox{(para }}b<0{\mbox{)}}\\\end{casos}}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\displaystyle \int x^{n}S\,dx={\frac {2}{a(2n+3)}}\left(x^{n}S^{3}-nb\int x^{ n-1}S\,dx\derecha)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
![{\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 )}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Referencias
- Abramowitz, Milton; Stegun, Irene A., eds. (1972). "Capítulo 3". Manual de funciones matemáticas con fórmulas, gráficas y tablas matemáticas . Nueva York: Dover.
- Gradshteyn, Izrail Solomonovich ; Ryzhik, Iosif Moiseevich ; Geronimus, Yuri Veniaminovich ; Tseytlin, Michail Yulyevich ; Jeffrey, Alan (2015) [octubre de 2014]. Zwillinger, Daniel; Moll, Víctor Hugo (eds.). Tabla de Integrales, Series y Productos . Traducido por Scripta Technica, Inc. (8 ed.). Prensa académica, Inc. ISBN 978-0-12-384933-5. LCCN 2014010276.(Varias ediciones anteriores también).
- Peirce, Benjamín Osgood (1929) [1899]. "Capítulo 3". Una breve tabla de integrales (3ª edición revisada). Boston: Ginn and Co. págs. 16-30.