С Википедије, слободне енциклопедије
Списак интеграла ирационалних функција :
∫
a
2
−
x
2
d
x
=
1
2
(
x
a
2
−
x
2
+
a
2
arcsin
x
a
)
(
|
x
|
≤
|
a
|
)
{\displaystyle \int {\sqrt {a^{2}-x^{2}}}\;dx={\frac {1}{2}}\left(x{\sqrt {a^{2}-x^{2}}}+a^{2}\arcsin {\frac {x}{a}}\right)\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}
∫
x
a
2
−
x
2
d
x
=
−
1
3
(
a
2
−
x
2
)
3
(
|
x
|
≤
|
a
|
)
{\displaystyle \int x{\sqrt {a^{2}-x^{2}}}\;dx=-{\frac {1}{3}}{\sqrt {(a^{2}-x^{2})^{3}}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}
∫
a
2
−
x
2
d
x
x
=
a
2
−
x
2
−
a
ln
|
a
+
a
2
+
x
2
x
|
(
|
x
|
≤
|
a
|
)
{\displaystyle \int {\frac {{\sqrt {a^{2}-x^{2}}}\;dx}{x}}={\sqrt {a^{2}-x^{2}}}-a\ln \left|{\frac {a+{\sqrt {a^{2}+x^{2}}}}{x}}\right|\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}
∫
d
x
a
2
−
x
2
=
arcsin
x
a
(
|
x
|
≤
|
a
|
)
{\displaystyle \int {\frac {dx}{\sqrt {a^{2}-x^{2}}}}=\arcsin {\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}
∫
x
2
d
x
a
2
−
x
2
=
−
x
2
a
2
−
x
2
+
a
2
2
arcsin
x
a
(
|
x
|
≤
|
a
|
)
{\displaystyle \int {\frac {x^{2}\;dx}{\sqrt {a^{2}-x^{2}}}}=-{\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}+{\frac {a^{2}}{2}}\arcsin {\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}}
∫
x
2
+
a
2
d
x
=
1
2
(
x
x
2
+
a
2
+
a
2
a
r
c
s
i
n
h
x
a
)
{\displaystyle \int {\sqrt {x^{2}+a^{2}}}\;dx={\frac {1}{2}}\left(x{\sqrt {x^{2}+a^{2}}}+a^{2}\,\mathrm {arcsinh} {\frac {x}{a}}\right)}
∫
x
x
2
+
a
2
d
x
=
1
3
(
x
2
+
a
2
)
3
{\displaystyle \int x{\sqrt {x^{2}+a^{2}}}\;dx={\frac {1}{3}}{\sqrt {(x^{2}+a^{2})^{3}}}}
∫
x
2
+
a
2
d
x
x
=
x
2
+
a
2
−
a
ln
|
a
+
x
2
+
a
2
x
|
{\displaystyle \int {\frac {{\sqrt {x^{2}+a^{2}}}\;dx}{x}}={\sqrt {x^{2}+a^{2}}}-a\ln \left|{\frac {a+{\sqrt {x^{2}+a^{2}}}}{x}}\right|}
∫
d
x
x
2
+
a
2
=
a
r
c
s
i
n
h
x
a
=
ln
|
x
+
x
2
+
a
2
|
{\displaystyle \int {\frac {dx}{\sqrt {x^{2}+a^{2}}}}=\mathrm {arcsinh} {\frac {x}{a}}=\ln \left|x+{\sqrt {x^{2}+a^{2}}}\right|}
∫
x
d
x
x
2
+
a
2
=
x
2
+
a
2
{\displaystyle \int {\frac {x\,dx}{\sqrt {x^{2}+a^{2}}}}={\sqrt {x^{2}+a^{2}}}}
∫
x
2
d
x
x
2
+
a
2
=
x
2
x
2
+
a
2
−
a
2
2
a
r
c
s
i
n
h
x
a
=
x
2
x
2
+
a
2
−
a
2
2
ln
|
x
+
x
2
+
a
2
|
{\displaystyle \int {\frac {x^{2}\;dx}{\sqrt {x^{2}+a^{2}}}}={\frac {x}{2}}{\sqrt {x^{2}+a^{2}}}-{\frac {a^{2}}{2}}\,\mathrm {arcsinh} {\frac {x}{a}}={\frac {x}{2}}{\sqrt {x^{2}+a^{2}}}-{\frac {a^{2}}{2}}\ln \left|x+{\sqrt {x^{2}+a^{2}}}\right|}
∫
d
x
x
x
2
+
a
2
=
−
1
a
a
r
c
s
i
n
h
a
x
=
−
1
a
ln
|
a
+
x
2
+
a
2
x
|
{\displaystyle \int {\frac {dx}{x{\sqrt {x^{2}+a^{2}}}}}=-{\frac {1}{a}}\,\mathrm {arcsinh} {\frac {a}{x}}=-{\frac {1}{a}}\ln \left|{\frac {a+{\sqrt {x^{2}+a^{2}}}}{x}}\right|}
∫
x
2
−
a
2
d
x
=
1
2
(
x
x
2
−
a
2
−
sgn
x
a
r
c
c
o
s
h
|
x
a
|
)
(za
|
x
|
≥
|
a
|
)
{\displaystyle \int {\sqrt {x^{2}-a^{2}}}\;dx={\frac {1}{2}}\left(x{\sqrt {x^{2}-a^{2}}}-\operatorname {sgn} x\,\mathrm {arccosh} \left|{\frac {x}{a}}\right|\right)\qquad {\mbox{(za }}|x|\geq |a|{\mbox{)}}}
∫
x
x
2
−
a
2
d
x
=
1
3
(
x
2
−
a
2
)
3
(za
|
x
|
≥
|
a
|
)
{\displaystyle \int x{\sqrt {x^{2}-a^{2}}}\;dx={\frac {1}{3}}{\sqrt {(x^{2}-a^{2})^{3}}}\qquad {\mbox{(za }}|x|\geq |a|{\mbox{)}}}
∫
x
2
−
a
2
d
x
x
=
x
2
−
a
2
−
a
arccos
a
x
(za
|
x
|
≥
|
a
|
)
{\displaystyle \int {\frac {{\sqrt {x^{2}-a^{2}}}\;dx}{x}}={\sqrt {x^{2}-a^{2}}}-a\arccos {\frac {a}{x}}\qquad {\mbox{(za }}|x|\geq |a|{\mbox{)}}}
∫
d
x
x
2
−
a
2
=
a
r
c
c
o
s
h
x
a
=
ln
(
|
x
|
+
x
2
−
a
2
)
(za
|
x
|
>
|
a
|
)
{\displaystyle \int {\frac {dx}{\sqrt {x^{2}-a^{2}}}}=\mathrm {arccosh} {\frac {x}{a}}=\ln \left(|x|+{\sqrt {x^{2}-a^{2}}}\right)\qquad {\mbox{(za }}|x|>|a|{\mbox{)}}}
∫
x
d
x
x
2
−
a
2
=
x
2
−
a
2
(za
|
x
|
>
|
a
|
)
{\displaystyle \int {\frac {x\;dx}{\sqrt {x^{2}-a^{2}}}}={\sqrt {x^{2}-a^{2}}}\qquad {\mbox{(za }}|x|>|a|{\mbox{)}}}
∫
x
2
d
x
x
2
−
a
2
=
x
2
x
2
−
a
2
+
a
2
2
a
r
c
c
o
s
h
|
x
a
|
=
1
2
(
x
x
2
−
a
2
+
a
2
ln
(
|
x
|
+
x
2
−
a
2
)
)
(za
|
x
|
>
|
a
|
)
{\displaystyle \int {\frac {x^{2}\,dx}{\sqrt {x^{2}-a^{2}}}}={\frac {x}{2}}{\sqrt {x^{2}-a^{2}}}+{\frac {a^{2}}{2}}\,\mathrm {arccosh} \left|{\frac {x}{a}}\right|={\frac {1}{2}}\left(x{\sqrt {x^{2}-a^{2}}}+a^{2}\ln \left(|x|+{\sqrt {x^{2}-a^{2}}}\right)\right)\qquad {\mbox{(za }}|x|>|a|{\mbox{)}}}
∫
a
x
2
+
b
x
+
c
d
x
=
1
8
a
3
/
2
(
2
a
⋅
(
b
+
2
a
x
)
c
+
x
(
b
+
a
x
)
−
(
b
2
−
4
a
c
)
log
(
b
+
2
x
+
2
a
c
+
x
(
b
+
a
x
)
)
)
{\displaystyle \int {{\sqrt {ax^{2}+bx+c}}\;dx}={\frac {1}{8a^{3/2}}}\left(2{\sqrt {a}}\cdot \left(b+2ax\right){\sqrt {c+x\left(b+ax\right)}}-\left(b^{2}-4ac\right)\log {\left(b+2x+2{\sqrt {a}}{\sqrt {c+x\left(b+ax\right)}}\right)}\right)}
∫
d
x
a
x
2
+
b
x
+
c
=
1
a
ln
|
2
a
(
a
x
2
+
b
x
+
c
)
+
2
a
x
+
b
|
(za
a
>
0
)
{\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\ln \left|2{\sqrt {a(ax^{2}+bx+c)}}+2ax+b\right|\qquad {\mbox{(za }}a>0{\mbox{)}}}
∫
d
x
a
x
2
+
b
x
+
c
=
1
a
a
r
c
s
i
n
h
2
a
x
+
b
4
a
c
−
b
2
(za
a
>
0
,
4
a
c
−
b
2
>
0
)
{\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\,\mathrm {arcsinh} {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{(za }}a>0{\mbox{, }}4ac-b^{2}>0{\mbox{)}}}
∫
d
x
a
x
2
+
b
x
+
c
=
1
a
ln
|
2
a
x
+
b
|
(za
a
>
0
,
4
a
c
−
b
2
=
0
)
{\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\ln |2ax+b|\qquad {\mbox{(za }}a>0{\mbox{, }}4ac-b^{2}=0{\mbox{)}}}
∫
d
x
a
x
2
+
b
x
+
c
=
−
1
−
a
arcsin
2
a
x
+
b
b
2
−
4
a
c
(za
a
<
0
,
4
a
c
−
b
2
<
0
)
{\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}=-{\frac {1}{\sqrt {-a}}}\arcsin {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad {\mbox{(za }}a<0{\mbox{, }}4ac-b^{2}<0{\mbox{)}}}
∫
x
d
x
a
x
2
+
b
x
+
c
=
a
x
2
+
b
x
+
c
a
−
b
2
a
∫
d
x
a
x
2
+
b
x
+
c
{\displaystyle \int {\frac {x\;dx}{\sqrt {ax^{2}+bx+c}}}={\frac {\sqrt {ax^{2}+bx+c}}{a}}-{\frac {b}{2a}}\int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}}
Milton Abramowitz and Irene Stegun, editors. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables .
I.S. Gradshteyn (И. С. Градштейн), I.M. Ryzhik (И. М. Рыжик); Alan Jeffrey, Daniel Zwillinger, ур. (2007). Table of Integrals, Series, and Products ISBN 978-0-12-373637-6 . Errata. (Several previous editions as well.) A.P. Prudnikov (А. П. Прудников), Yu.A. Brychkov (Ю. А. Брычков), O.I. Marichev (О. И. Маричев). Integrals and Series . First edition (Russian), volume 1–5, Nauka , 1981−1986. First edition (English, translated from the Russian by N.M. Queen), volume 1–5, Gordon & Breach Science Publishers/CRC Press, 1988–1992. ISBN 978-2-88124-097-3 .. Second revised edition (Russian), volume 1–3, Fiziko-Matematicheskaya Literatura, 2003.
Yu.A. Brychkov (Ю. А. Брычков). Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas . Russian edition, Fiziko-Matematicheskaya Literatura, 2006. English edition. . Chapman & Hall/CRC Press. 2008. ISBN 978-1-58488-956-4 .
Zwillinger, Daniel (2002). CRC Standard Mathematical Tables and Formulae . 31st edition. Chapman & Hall/CRC Press. ISBN 978-1-58488-291-6 . (Many earlier editions as well.) Meyer Hirsch, Integral Tables, Or, A Collection of Integral Formulae (Baynes and son, London, 1823) [English translation of Integraltafeln ]
Benjamin O. Pierce A short table of integrals - revised edition (Ginn & co., Boston, 1899)
