Lijst van integralen van irrationale functies

Dit artikel bevat een lijst van integralen van irrationale functies.

Integralen met $r={\sqrt {x^{2}+a^{2}}}$

$\int r\,\mathrm {d} x={\tfrac {1}{2}}(xr+a^{2}\,\ln(x+r))$

$\int r^{3}\,\mathrm {d} x={\tfrac {1}{4}}xr^{3}+{\tfrac {3}{8}}a^{2}xr+{\tfrac {3}{8}}a^{4}\ln(x+r)$

$\int r^{5}\,\mathrm {d} x={\tfrac {1}{6}}xr^{5}+{\tfrac {5}{24}}a^{2}xr^{3}+{\tfrac {5}{16}}a^{4}xr+{\tfrac {5}{16}}a^{6}\ln(x+r)$

$\int xr\,\mathrm {d} x={\tfrac {1}{3}}r^{3}$

$\int xr^{3}\,\mathrm {d} x={\tfrac {1}{5}}r^{5}$

$\int xr^{2n+1}\,\mathrm {d} x={\frac {r^{2n+3}}{2n+3}}$

$\int x^{2}r\,\mathrm {d} x={\tfrac {1}{4}}xr^{3}-{\tfrac {1}{8}}a^{2}xr-{\tfrac {1}{8}}a^{4}\ln(x+r)$

$\int x^{2}r^{3}\,\mathrm {d} x={\tfrac {1}{6}}xr^{5}-{\tfrac {1}{24}}a^{2}xr^{3}-{\tfrac {1}{16}}a^{4}xr-{\tfrac {1}{16}}a^{6}\ln(x+r)$

$\int x^{3}r\,\mathrm {d} x={\tfrac {1}{5}}r^{5}-{\tfrac {1}{3}}a^{2}r^{3}$

$\int x^{3}r^{3}\,\mathrm {d} x={\tfrac {1}{7}}r^{7}-{\tfrac {1}{5}}a^{2}r^{5}$

$\int x^{3}r^{2n+1}\,\mathrm {d} x={\frac {r^{2n+5}}{2n+5}}-{\frac {a^{3}r^{2n+3}}{2n+3}}$

$\int x^{4}r\,\mathrm {d} x={\tfrac {1}{6}}x^{3}r^{3}-{\tfrac {1}{8}}a^{2}xr^{3}+{\tfrac {1}{16}}a^{4}xr+{\tfrac {1}{16}}a^{6}\ln(x+r)$

$\int x^{4}r^{3}\,\mathrm {d} x={\tfrac {1}{8}}x^{3}r^{5}-{\tfrac {1}{16}}a^{2}xr^{5}+{\tfrac {1}{64}}a^{4}xr^{3}+{\tfrac {3}{128}}a^{6}xr+{\tfrac {3}{128}}a^{8}\ln(x+r)$

$\int x^{5}r\,\mathrm {d} x={\tfrac {1}{7}}r^{7}-{\tfrac {2}{5}}a^{2}r^{5}+{\tfrac {1}{3}}a^{4}r^{3}$

$\int x^{5}r^{3}\,\mathrm {d} x={\tfrac {1}{9}}r^{9}-{\tfrac {2}{7}}a^{2}r^{7}+{\tfrac {1}{5}}a^{4}r^{5}$

$\int x^{5}r^{2n+1}\,\mathrm {d} x={\frac {r^{2n+7}}{2n+7}}-{\frac {2a^{2}r^{2n+5}}{2n+5}}+{\frac {a^{4}r^{2n+3}}{2n+3}}$

$\int {\frac {r\,\mathrm {d} x}{x}}=r-a\ln \left|{\frac {a+r}{x}}\right|=r-a\,\operatorname {arsinh} {\frac {a}{x}}$

$\int {\frac {r^{3}\,\mathrm {d} x}{x}}={\tfrac {1}{3}}r^{3}+a^{2}r-a^{3}\ln \left|{\frac {a+r}{x}}\right|$

$\int {\frac {r^{5}\,\mathrm {d} x}{x}}={\tfrac {1}{5}}r^{5}+{\tfrac {1}{3}}a^{2}r^{3}+a^{4}r-a^{5}\ln \left|{\frac {a+r}{x}}\right|$

$\int {\frac {r^{7}\,\mathrm {d} x}{x}}={\tfrac {1}{7}}r^{7}+{\tfrac {1}{5}}a^{2}r^{5}+{\tfrac {1}{3}}a^{4}r^{3}+a^{6}r-a^{7}\ln \left|{\frac {a+r}{x}}\right|$

$\int {\frac {\mathrm {d} x}{r}}=\operatorname {arsinh} {\frac {x}{a}}=\ln \left({\frac {x+r}{a}}\right)$

$\int {\frac {\mathrm {d} x}{r^{3}}}={\frac {x}{a^{2}r}}$

$\int {\frac {x\,\,\mathrm {d} x}{r}}=r$

$\int {\frac {x\,\,\mathrm {d} x}{r^{3}}}=-{\frac {1}{r}}$

$\int {\frac {x^{2}\,\mathrm {d} x}{r}}={\tfrac {1}{2}}xr-{\tfrac {1}{2}}a^{2}\,\operatorname {arsinh} {\frac {x}{a}}={\tfrac {1}{2}}xr-{\tfrac {1}{2}}a^{2}\ln \left({\frac {x+r}{a}}\right)$

$\int {\frac {\mathrm {d} x}{xr}}=-{\frac {1}{a}}\,\operatorname {arsinh} {\frac {a}{x}}=-{\frac {1}{a}}\ln \left|{\frac {a+r}{x}}\right|$

Integralen met $s={\sqrt {x^{2}-a^{2}}}$

Neem aan dat $x^{2}>a^{2}$, voor de situatie $x^{2}<a^{2}$, zie de volgende paragraaf.

$\int s\,\mathrm {d} x={\tfrac {1}{2}}(xs-a^{2}\ln(x+s))$

$\int xs\,\mathrm {d} x={\tfrac {1}{3}}s^{3}$

$\int {\frac {s\,\mathrm {d} x}{x}}=s-a\arccos \left|{\frac {a}{x}}\right|$

$\int {\frac {\mathrm {d} x}{s}}=\int {\frac {\mathrm {d} x}{\sqrt {x^{2}-a^{2}}}}=\ln \left|{\frac {x+s}{a}}\right|$

Houd hier rekening met het feit dat $\ln \left|{\frac {x+s}{a}}\right|=\mathrm {sgn} (x)\,\operatorname {arcosh} \left|{\frac {x}{a}}\right|={\frac {1}{2}}\ln \left({\frac {x+s}{x-s}}\right)$, waarbij enkel moet gekeken worden naar de positieve waarde, namelijk $\operatorname {arcosh} \left|{\frac {x}{a}}\right|$

$\int {\frac {x\,\mathrm {d} x}{s}}=s$

$\int {\frac {x\,\mathrm {d} x}{s^{3}}}=-{\frac {1}{s}}$

$\int {\frac {x\,\mathrm {d} x}{s^{5}}}=-{\frac {1}{3s^{3}}}$

$\int {\frac {x\,\mathrm {d} x}{s^{7}}}=-{\frac {1}{5s^{5}}}$

$\int {\frac {x\,\mathrm {d} x}{s^{2n+1}}}=-{\frac {1}{(2n-1)s^{2n-1}}}$

$\int {\frac {x^{2m}\,\mathrm {d} x}{s^{2n+1}}}=-{\frac {1}{2n-1}}{\frac {x^{2m-1}}{s^{2n-1}}}+{\frac {2m-1}{2n-1}}\int {\frac {x^{2m-2}\,\mathrm {d} x}{s^{2n-1}}}$

$\int {\frac {x^{2}\,\mathrm {d} x}{s}}={\tfrac {1}{2}}xs+{\tfrac {1}{2}}a^{2}\ln \left|{\frac {x+s}{a}}\right|$

$\int {\frac {x^{2}\,\mathrm {d} x}{s^{3}}}=-{\frac {x}{s}}+\ln \left|{\frac {x+s}{a}}\right|$

$\int {\frac {x^{4}\,\mathrm {d} x}{s}}={\tfrac {1}{4}}x^{3}s+{\tfrac {3}{8}}a^{2}xs+{\tfrac {3}{8}}a^{4}\ln \left|{\frac {x+s}{a}}\right|$

$\int {\frac {x^{4}\,\mathrm {d} x}{s^{3}}}={\tfrac {1}{2}}xs-{\frac {a^{2}x}{s}}+{\tfrac {3}{2}}a^{2}\ln \left|{\frac {x+s}{a}}\right|$

$\int {\frac {x^{4}\,\mathrm {d} x}{s^{5}}}=-{\frac {x}{s}}-{\frac {x^{3}}{3s^{3}}}+\ln \left|{\frac {x+s}{a}}\right|$

$\int {\frac {x^{2m}\,\mathrm {d} x}{s^{2n+1}}}=(-1)^{n-m}{\frac {1}{a^{2(n-m)}}}\sum _{i=0}^{n-m-1}{\frac {1}{2(m+i)+1}}{n-m-1 \choose i}{\frac {x^{2(m+i)+1}}{s^{2(m+i)+1}}}\qquad (n>m\geq 0)$

$\int {\frac {\mathrm {d} x}{s^{3}}}=-{\frac {1}{a^{2}}}{\frac {x}{s}}$

$\int {\frac {\mathrm {d} x}{s^{5}}}={\frac {1}{a^{4}}}\left[{\frac {x}{s}}-{\tfrac {1}{3}}{\frac {x^{3}}{s^{3}}}\right]$

$\int {\frac {\mathrm {d} x}{s^{7}}}=-{\frac {1}{a^{6}}}\left[{\frac {x}{s}}-{\tfrac {2}{3}}{\frac {x^{3}}{s^{3}}}+{\tfrac {1}{5}}{\frac {x^{5}}{s^{5}}}\right]$

$\int {\frac {\mathrm {d} x}{s^{9}}}={\frac {1}{a^{8}}}\left[{\frac {x}{s}}-{\tfrac {3}{3}}{\frac {x^{3}}{s^{3}}}+{\tfrac {3}{5}}{\frac {x^{5}}{s^{5}}}-{\tfrac {1}{7}}{\frac {x^{7}}{s^{7}}}\right]$

$\int {\frac {x^{2}\,\mathrm {d} x}{s^{5}}}=-{\frac {1}{a^{2}}}{\frac {x^{3}}{3s^{3}}}$

$\int {\frac {x^{2}\,\mathrm {d} x}{s^{7}}}={\frac {1}{a^{4}}}\left[{\tfrac {1}{3}}{\frac {x^{3}}{3s^{3}}}-{\tfrac {1}{5}}{\frac {x^{5}}{s^{5}}}\right]$

$\int {\frac {x^{2}\,\mathrm {d} x}{s^{9}}}=-{\frac {1}{a^{6}}}\left[{\tfrac {1}{3}}{\frac {x^{3}}{s^{3}}}-{\tfrac {2}{5}}{\frac {x^{5}}{s^{5}}}+{\tfrac {1}{7}}{\frac {x^{7}}{s^{7}}}\right]$

Integralen waarbij $u={\sqrt {a^{2}-x^{2}}}$

$\int u\,\mathrm {d} x={\frac {1}{2}}\left(xu+a^{2}\arcsin {\frac {x}{a}}\right)\qquad (|x|\leq |a|)$

$\int xu\,\mathrm {d} x=-{\frac {1}{3}}u^{3}\qquad (|x|\leq |a|)$

$\int x^{2}u\,\mathrm {d} x=-{\frac {x}{4}}u^{3}+{\frac {a^{2}}{8}}(xu+a^{2}\arcsin {\frac {x}{a}})\qquad (|x|\leq |a|)$

$\int {\frac {u\,\mathrm {d} x}{x}}=u-a\ln \left|{\frac {a+u}{x}}\right|\qquad (|x|\leq |a|)$

$\int {\frac {\mathrm {d} x}{u}}=\arcsin {\frac {x}{a}}\qquad (|x|\leq |a|)$

$\int {\frac {x^{2}\,\mathrm {d} x}{u}}={\frac {1}{2}}\left(-xu+a^{2}\arcsin {\frac {x}{a}}\right)\qquad (|x|\leq |a|)$

$\int u\,\mathrm {d} x={\frac {1}{2}}\left(xu-\operatorname {sgn} x\,\operatorname {arcosh} \left|{\frac {x}{a}}\right|\right)\qquad ({\mbox{voor }}|x|\geq |a|)$

$\int {\frac {x}{u}}\,\mathrm {d} x=-u\qquad (|x|\leq |a|)$

Integralen waarbij $R={\sqrt {ax^{2}+bx+c}}$

Neem aan dat $ax^{2}+bx+c$ niet geschrevenkan worden als de verkorte vorm $(px+q)^{2}$

$\int {\frac {\mathrm {d} x}{R}}={\frac {1}{\sqrt {a}}}\ln \left|2{\sqrt {a}}R+2ax+b\right|\qquad ({\mbox{voor }}a>0)$

$\int {\frac {\mathrm {d} x}{R}}={\frac {1}{\sqrt {a}}}\,\operatorname {arsinh} {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad ({\mbox{voor }}a>0,\ 4ac-b^{2}>0)$

$\int {\frac {\mathrm {d} x}{R}}={\frac {1}{\sqrt {a}}}\ln |2ax+b|\quad ({\mbox{voor }}a>0,\ 4ac-b^{2}=0)$

$\int {\frac {\mathrm {d} x}{R}}=-{\frac {1}{\sqrt {-a}}}\arcsin {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad ({\mbox{voor }}a<0,\ 4ac-b^{2}<0,\ |2ax+b|<{\sqrt {b^{2}-4ac}})$

$\int {\frac {\mathrm {d} x}{R^{3}}}={\frac {4ax+2b}{(4ac-b^{2})R}}$

$\int {\frac {\mathrm {d} x}{R^{5}}}={\frac {4ax+2b}{3(4ac-b^{2})R}}\left({\frac {1}{R^{2}}}+{\frac {8a}{4ac-b^{2}}}\right)$

$\int {\frac {\mathrm {d} x}{R^{2n+1}}}={\frac {2}{(2n-1)(4ac-b^{2})}}\left({\frac {2ax+b}{R^{2n-1}}}+4a(n-1)\int {\frac {\mathrm {d} x}{R^{2n-1}}}\right)$

$\int {\frac {x}{R}}\,\mathrm {d} x={\frac {R}{a}}-{\frac {b}{2a}}\int {\frac {\mathrm {d} x}{R}}$

$\int {\frac {x}{R^{3}}}\,\mathrm {d} x=-{\frac {2bx+4c}{(4ac-b^{2})R}}$

$\int {\frac {x}{R^{2n+1}}}\,\mathrm {d} x=-{\frac {1}{(2n-1)aR^{2n-1}}}-{\frac {b}{2a}}\int {\frac {\mathrm {d} x}{R^{2n+1}}}$

$\int {\frac {\mathrm {d} x}{xR}}=-{\frac {1}{\sqrt {c}}}\ln \left({\frac {2{\sqrt {c}}R+bx+2c}{x}}\right)$

$\int {\frac {\mathrm {d} x}{xR}}=-{\frac {1}{\sqrt {c}}}\operatorname {arsinh} \left({\frac {bx+2c}{|x|{\sqrt {4ac-b^{2}}}}}\right)$

Integralen waarbij $S={\sqrt {ax+b}}$

$\int S{\mathrm {d} x}={\frac {2S^{3}}{3a}}$

$\int {\frac {\mathrm {d} x}{S}}={\frac {2S}{a}}$

$\int {\frac {\mathrm {d} x}{xS}}={\begin{cases}({\mbox{voor }}b>0,\quad ax>0)\\-{\frac {2}{\sqrt {b}}}\mathrm {artanh} \left({\frac {S}{\sqrt {b}}}\right)&({\mbox{voor }}b>0,\quad ax<0)\\{\frac {2}{\sqrt {-b}}}\arctan \left({\frac {S}{\sqrt {-b}}}\right)&({\mbox{voor }}b<0)\\\end{cases}}$

$\int {\frac {S}{x}}\,\mathrm {d} x={\begin{cases}2\left(S-{\sqrt {b}}\,\mathrm {arcoth} \left({\frac {S}{\sqrt {b}}}\right)\right)&{\mbox{(voor }}b>0,\quad ax>0{\mbox{)}}\\2\left(S-{\sqrt {b}}\,\mathrm {artanh} \left({\frac {S}{\sqrt {b}}}\right)\right)&{\mbox{(voor }}b>0,\quad ax<0{\mbox{)}}\\2\left(S-{\sqrt {-b}}\arctan \left({\frac {S}{\sqrt {-b}}}\right)\right)&{\mbox{(voor }}b<0{\mbox{)}}\\\end{cases}}$

$\int {\frac {x^{n}}{S}}\,\mathrm {d} x={\frac {2}{a(2n+1)}}\left(x^{n}S-bn\int {\frac {x^{n-1}}{S}}\,\mathrm {d} x\right)$

$\int x^{n}S\,\mathrm {d} x={\frac {2}{a(2n+3)}}\left(x^{n}S^{3}-nb\int x^{n-1}S\,\mathrm {d} x\right)$

$\int {\frac {1}{x^{n}S}}\,\mathrm {d} x=-{\frac {1}{b(n-1)}}\left({\frac {S}{x^{n-1}}}+\left(n-{\frac {3}{2}}\right)a\int {\frac {\mathrm {d} x}{x^{n-1}S}}\right)$

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Lijst van integralen

rationaal · irrationaal · exponentieel · logaritmisch · goniometrisch · invers goniometrisch · hyperbolisch · invers hyperbolisch

Bronnen, noten en/of referenties

• Milton Abramowitz & Irene A. Stegun - Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables 1972, Dover: New York
• S. Gradshteyn, I.M. Ryzhik, Alan Jeffrey, Daniel Zwillinger - Table of Integrals, Series, and Products, seventh edition. Academic Press, 2007 - ISBN 978-0-12-373637-6.