Seznam integrálů iracionálních funkcí

Toto je seznam integrálů (primitivních funkcí) iracionálních funkcí.

Integrály s $r={\sqrt {x^{2} a^{2}}}$

$\int r\;\mathrm {d} x={\frac {1}{2}}\left[xr a^{2}\,\ln \left({\frac {x r}{a}}\right)\right]$

$\int r^{3}\;\mathrm {d} x={\frac {1}{4}}xr^{3} {\frac {1}{8}}3a^{2}xr {\frac {3}{8}}a^{4}\ln \left({\frac {x r}{a}}\right)$

$\int r^{5}\;\mathrm {d} x={\frac {1}{6}}xr^{5} {\frac {5}{24}}a^{2}xr^{3} {\frac {5}{16}}a^{4}xr {\frac {5}{16}}a^{6}\ln \left({\frac {x r}{a}}\right)$

$\int xr\;\mathrm {d} x={\frac {r^{3}}{3}}$

$\int xr^{3}\;\mathrm {d} x={\frac {r^{5}}{5}}$

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

$\int x^{2}r\;\mathrm {d} x={\frac {xr^{3}}{4}}-{\frac {a^{2}xr}{8}}-{\frac {a^{4}}{8}}\ln \left({\frac {x r}{a}}\right)$

$\int x^{2}r^{3}\;\mathrm {d} x={\frac {xr^{5}}{6}}-{\frac {a^{2}xr^{3}}{24}}-{\frac {a^{4}xr}{16}}-{\frac {a^{6}}{16}}\ln \left({\frac {x r}{a}}\right)$

$\int x^{3}r\;\mathrm {d} x={\frac {r^{5}}{5}}-{\frac {a^{2}r^{3}}{3}}$

$\int x^{3}r^{3}\;\mathrm {d} x={\frac {r^{7}}{7}}-{\frac {a^{2}r^{5}}{5}}$

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

$\int x^{4}r\;\mathrm {d} x={\frac {x^{3}r^{3}}{6}}-{\frac {a^{2}xr^{3}}{8}} {\frac {a^{4}xr}{16}} {\frac {a^{6}}{16}}\ln \left({\frac {x r}{a}}\right)$

$\int x^{4}r^{3}\;\mathrm {d} x={\frac {x^{3}r^{5}}{8}}-{\frac {a^{2}xr^{5}}{16}} {\frac {a^{4}xr^{3}}{64}} {\frac {3a^{6}xr}{128}} {\frac {3a^{8}}{128}}\ln \left({\frac {x r}{a}}\right)$

$\int x^{5}r\;\mathrm {d} x={\frac {r^{7}}{7}}-{\frac {2a^{2}r^{5}}{5}} {\frac {a^{4}r^{3}}{3}}$

$\int x^{5}r^{3}\;\mathrm {d} x={\frac {r^{9}}{9}}-{\frac {2a^{2}r^{7}}{7}} {\frac {a^{4}r^{5}}{5}}$

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

$\int {\frac {r\;\mathrm {d} x}{x}}=r-a\ln \left|{\frac {a r}{x}}\right|=r-a\sinh ^{-1}{\frac {a}{x}}$

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

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

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

$\int {\frac {\mathrm {d} x}{r}}=\sinh ^{-1}{\frac {x}{a}}=\ln \left|x r\right|$

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

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

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

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

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

Integrály s $s={\sqrt {x^{2}-a^{2}}}$

Předpokládejme $(x^{2}>a^{2})$, pro $(x^{2}<a^{2})$, viz další sekce:

$\int xs\;\mathrm {d} x={\frac {1}{3}}s^{3}$

$\int {\frac {s\;\mathrm {d} x}{x}}=s-a\cos ^{-1}\left|{\frac {a}{x}}\right|$

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

Poznámka: $\ln \left|{\frac {x s}{a}}\right|=\mathrm {sgn} (x)\cosh ^{-1}\left|{\frac {x}{a}}\right|={\frac {1}{2}}\ln \left({\frac {x s}{x-s}}\right)$, kde vezmeme kladnou hodnotu $\cosh ^{-1}\left|{\frac {x}{a}}\right|$.

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

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

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

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

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

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

$\int {\frac {x^{2}\;\mathrm {d} x}{s}}={\frac {xs}{2}} {\frac {a^{2}}{2}}\ln \left|{\frac {x s}{a}}\right|$

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

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

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

$\int {\frac {x^{4}\;\mathrm {d} x}{s^{5}}}=-{\frac {x}{s}}-{\frac {1}{3}}{\frac {x^{3}}{s^{3}}} \ln \left|{\frac {x s}{a}}\right|$

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

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

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

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

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

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

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

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

Integrály s $t={\sqrt {a^{2}-x^{2}}}$

$\int t\;\mathrm {d} x={\frac {1}{2}}\left(xt a^{2}\sin ^{-1}{\frac {x}{a}}\right)\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}$

$\int xt\;\mathrm {d} x=-{\frac {1}{3}}t^{3}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}$

$\int {\frac {t\;\mathrm {d} x}{x}}=t-a\ln \left|{\frac {a t}{x}}\right|\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}$

$\int {\frac {\mathrm {d} x}{t}}=\sin ^{-1}{\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}$

$\int {\frac {x^{2}\;\mathrm {d} x}{t}}=-{\frac {x}{2}}t {\frac {a^{2}}{2}}\sin ^{-1}{\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}$

$\int t\;\mathrm {d} x={\frac {1}{2}}\left(xt-\operatorname {sgn} x\,\cosh ^{-1}\left|{\frac {x}{a}}\right|\right)\qquad {\mbox{(pro }}|x|\geq |a|{\mbox{)}}$

Integrály s $R^{1/2}={\sqrt {ax^{2} bx c}}$

$\int {\frac {\mathrm {d} x}{\sqrt {ax^{2} bx c}}}={\frac {1}{\sqrt {a}}}\ln \left|2{\sqrt {aR}} 2ax b\right|\qquad {\mbox{(pro }}a>0{\mbox{)}}$

$\int {\frac {\mathrm {d} x}{\sqrt {ax^{2} bx c}}}={\frac {1}{\sqrt {a}}}\,\sinh ^{-1}{\frac {2ax b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{(pro }}a>0{\mbox{, }}4ac-b^{2}>0{\mbox{)}}$

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

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

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

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

$\int {\frac {\mathrm {d} x}{\sqrt {(ax^{2} bx c)^{2n 1}}}}={\frac {4ax 2b}{(2n-1)(4ac-b^{2})R^{(2n-1)/2}}} {\frac {8a(n-1)}{(2n-1)(4ac-b^{2})}}\int {\frac {\mathrm {d} x}{R^{(2n-1)/2}}}$

$\int {\frac {x\;\mathrm {d} x}{\sqrt {ax^{2} bx c}}}={\frac {\sqrt {R}}{a}}-{\frac {b}{2a}}\int {\frac {\mathrm {d} x}{\sqrt {R}}}$

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

$\int {\frac {x\;\mathrm {d} x}{\sqrt {(ax^{2} bx c)^{2n 1}}}}=-{\frac {1}{(2n-1)aR^{(2n-1)/2}}}-{\frac {b}{2a}}\int {\frac {\mathrm {d} x}{R^{(2n 1)/2}}}$

$\int {\frac {\mathrm {d} x}{x{\sqrt {ax^{2} bx c}}}}=-{\frac {1}{\sqrt {c}}}\ln \left({\frac {2{\sqrt {cR}} bx 2c}{x}}\right)$

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

Integrály s $R^{1/2}={\sqrt {ax b}}$

$\int {\frac {\mathrm {d} x}{x{\sqrt {ax b}}}}\,=\,{\frac {-2}{\sqrt {b}}}\tanh ^{-1}{\sqrt {\frac {ax b}{b}}}$

$\int {\frac {\sqrt {ax b}}{x}}\,\mathrm {d} x\;=\;2\left({\sqrt {ax b}}-{\sqrt {b}}\tanh ^{-1}{\sqrt {\frac {ax b}{b}}}\right)$

$\int {\frac {x^{n}}{\sqrt {ax b}}}\,\mathrm {d} x\;=\;{\frac {2}{a(2n 1)}}\left(x^{n}{\sqrt {ax b}}-bn\int {\frac {x^{n-1}}{\sqrt {ax b}}}\right)$

$\int x^{n}{\sqrt {ax b}}\,\mathrm {d} x\;=\;{\frac {2}{2n 1}}\left(x^{n 1}{\sqrt {ax b}} bx^{n}{\sqrt {ax b}}-nb\int x^{n-1}{\sqrt {ax b}}\,\mathrm {d} x\right)$