Seznam integrálů racionálních funkcí
Toto je seznam integrálů (primitivních funkcí) racionálních funkcí .
∫
(
a
x
+
b
)
n
d
x
{\displaystyle \int (ax+b)^{n}\mathrm {d} x}
=
(
a
x
+
b
)
n
+
1
a
(
n
+
1
)
(pro
n
≠
−
1
)
{\displaystyle ={\frac {(ax+b)^{n+1}{a(n+1)}\qquad {\mbox{(pro }n\neq -1{\mbox{)}\,\!}
∫
d
x
a
x
+
b
{\displaystyle \int {\frac {\mathrm {d} x}{ax+b}
=
1
a
ln
|
a
x
+
b
|
{\displaystyle ={\frac {1}{a}\ln \left|ax+b\right|}
∫
x
(
a
x
+
b
)
n
d
x
{\displaystyle \int x(ax+b)^{n}\mathrm {d} x}
=
a
(
n
+
1
)
x
−
b
a
2
(
n
+
1
)
(
n
+
2
)
(
a
x
+
b
)
n
+
1
(pro
n
∉
{
−
1
,
−
2
}
)
{\displaystyle ={\frac {a(n+1)x-b}{a^{2}(n+1)(n+2)}(ax+b)^{n+1}\qquad {\mbox{(pro }n\not \in \{-1,-2\}{\mbox{)}
∫
x
d
x
a
x
+
b
{\displaystyle \int {\frac {x\,\mathrm {d} x}{ax+b}
=
x
a
−
b
a
2
ln
|
a
x
+
b
|
{\displaystyle ={\frac {x}{a}-{\frac {b}{a^{2}\ln \left|ax+b\right|}
∫
x
d
x
(
a
x
+
b
)
2
{\displaystyle \int {\frac {x\,\mathrm {d} x}{(ax+b)^{2}
=
b
a
2
(
a
x
+
b
)
+
1
a
2
ln
|
a
x
+
b
|
{\displaystyle ={\frac {b}{a^{2}(ax+b)}+{\frac {1}{a^{2}\ln \left|ax+b\right|}
∫
x
d
x
(
a
x
+
b
)
n
{\displaystyle \int {\frac {x\,\mathrm {d} x}{(ax+b)^{n}
=
a
(
1
−
n
)
x
−
b
a
2
(
n
−
1
)
(
n
−
2
)
(
a
x
+
b
)
n
−
1
(pro
n
∉
{
1
,
2
}
)
{\displaystyle ={\frac {a(1-n)x-b}{a^{2}(n-1)(n-2)(ax+b)^{n-1}\qquad {\mbox{(pro }n\not \in \{1,2\}{\mbox{)}
∫
x
2
d
x
a
x
+
b
{\displaystyle \int {\frac {x^{2}\mathrm {d} x}{ax+b}
=
1
a
3
(
(
a
x
+
b
)
2
2
−
2
b
(
a
x
+
b
)
+
b
2
ln
|
a
x
+
b
|
)
{\displaystyle ={\frac {1}{a^{3}\left({\frac {(ax+b)^{2}{2}-2b(ax+b)+b^{2}\ln \left|ax+b\right|\right)}
∫
x
2
d
x
(
a
x
+
b
)
2
{\displaystyle \int {\frac {x^{2}\mathrm {d} x}{(ax+b)^{2}
=
1
a
3
(
a
x
+
b
−
2
b
ln
|
a
x
+
b
|
−
b
2
a
x
+
b
)
{\displaystyle ={\frac {1}{a^{3}\left(ax+b-2b\ln \left|ax+b\right|-{\frac {b^{2}{ax+b}\right)}
∫
x
2
d
x
(
a
x
+
b
)
3
{\displaystyle \int {\frac {x^{2}\mathrm {d} x}{(ax+b)^{3}
=
1
a
3
(
ln
|
a
x
+
b
|
+
2
b
a
x
+
b
−
b
2
2
(
a
x
+
b
)
2
)
{\displaystyle ={\frac {1}{a^{3}\left(\ln \left|ax+b\right|+{\frac {2b}{ax+b}-{\frac {b^{2}{2(ax+b)^{2}\right)}
∫
x
2
d
x
(
a
x
+
b
)
n
{\displaystyle \int {\frac {x^{2}\mathrm {d} x}{(ax+b)^{n}
=
1
a
3
(
−
(
a
x
+
b
)
3
−
n
(
n
−
3
)
+
2
b
(
a
+
b
)
2
−
n
(
n
−
2
)
−
b
2
(
a
x
+
b
)
1
−
n
(
n
−
1
)
)
(pro
n
∉
{
1
,
2
,
3
}
)
{\displaystyle ={\frac {1}{a^{3}\left(-{\frac {(ax+b)^{3-n}{(n-3)}+{\frac {2b(a+b)^{2-n}{(n-2)}-{\frac {b^{2}(ax+b)^{1-n}{(n-1)}\right)\qquad {\mbox{(pro }n\not \in \{1,2,3\}{\mbox{)}
∫
d
x
x
(
a
x
+
b
)
{\displaystyle \int {\frac {\mathrm {d} x}{x(ax+b)}
=
−
1
b
ln
|
a
x
+
b
x
|
{\displaystyle =-{\frac {1}{b}\ln \left|{\frac {ax+b}{x}\right|}
∫
d
x
x
2
(
a
x
+
b
)
{\displaystyle \int {\frac {\mathrm {d} x}{x^{2}(ax+b)}
=
−
1
b
x
+
a
b
2
ln
|
a
x
+
b
x
|
{\displaystyle =-{\frac {1}{bx}+{\frac {a}{b^{2}\ln \left|{\frac {ax+b}{x}\right|}
∫
d
x
x
2
(
a
x
+
b
)
2
{\displaystyle \int {\frac {\mathrm {d} x}{x^{2}(ax+b)^{2}
=
−
a
(
1
b
2
(
a
x
+
b
)
+
1
a
b
2
x
−
2
b
3
ln
|
a
x
+
b
x
|
)
{\displaystyle =-a\left({\frac {1}{b^{2}(ax+b)}+{\frac {1}{ab^{2}x}-{\frac {2}{b^{3}\ln \left|{\frac {ax+b}{x}\right|\right)}
∫
d
x
x
2
+
a
2
{\displaystyle \int {\frac {\mathrm {d} x}{x^{2}+a^{2}
=
1
a
arctan
x
a
{\displaystyle ={\frac {1}{a}\arctan {\frac {x}{a}\,\!}
∫
d
x
x
2
−
a
2
=
{\displaystyle \int {\frac {\mathrm {d} x}{x^{2}-a^{2}=}
−
1
a
a
r
c
t
a
n
h
x
a
=
1
2
a
ln
a
−
x
a
+
x
(pro
|
x
|
<
|
a
|
)
{\displaystyle -{\frac {1}{a}\,\mathrm {arctanh} {\frac {x}{a}={\frac {1}{2a}\ln {\frac {a-x}{a+x}\qquad {\mbox{(pro }|x|<|a|{\mbox{)}\,\!}
−
1
a
a
r
c
c
o
t
h
x
a
=
1
2
a
ln
x
−
a
x
+
a
(pro
|
x
|
>
|
a
|
)
{\displaystyle -{\frac {1}{a}\,\mathrm {arccoth} {\frac {x}{a}={\frac {1}{2a}\ln {\frac {x-a}{x+a}\qquad {\mbox{(pro }|x|>|a|{\mbox{)}\,\!}
∫
d
x
a
x
2
+
b
x
+
c
=
{\displaystyle \int {\frac {\mathrm {d} x}{ax^{2}+bx+c}=}
2
4
a
c
−
b
2
arctan
2
a
x
+
b
4
a
c
−
b
2
(pro
4
a
c
−
b
2
>
0
)
{\displaystyle {\frac {2}{\sqrt {4ac-b^{2}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}\qquad {\mbox{(pro }4ac-b^{2}>0{\mbox{)}
2
b
2
−
4
a
c
a
r
t
a
n
h
2
a
x
+
b
b
2
−
4
a
c
=
1
b
2
−
4
a
c
ln
|
2
a
x
+
b
−
b
2
−
4
a
c
2
a
x
+
b
+
b
2
−
4
a
c
|
(pro
4
a
c
−
b
2
<
0
)
{\displaystyle {\frac {2}{\sqrt {b^{2}-4ac}\,\mathrm {artanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}={\frac {1}{\sqrt {b^{2}-4ac}\ln \left|{\frac {2ax+b-{\sqrt {b^{2}-4ac}{2ax+b+{\sqrt {b^{2}-4ac}\right|\qquad {\mbox{(pro }4ac-b^{2}<0{\mbox{)}
−
2
2
a
x
+
b
(pro
4
a
c
−
b
2
=
0
)
{\displaystyle -{\frac {2}{2ax+b}\qquad {\mbox{(pro }4ac-b^{2}=0{\mbox{)}
∫
x
d
x
a
x
2
+
b
x
+
c
{\displaystyle \int {\frac {x\,\mathrm {d} x}{ax^{2}+bx+c}
=
1
2
a
ln
|
a
x
2
+
b
x
+
c
|
−
b
2
a
∫
d
x
a
x
2
+
b
x
+
c
{\displaystyle ={\frac {1}{2a}\ln \left|ax^{2}+bx+c\right|-{\frac {b}{2a}\int {\frac {\mathrm {d} x}{ax^{2}+bx+c}
∫
(
m
x
+
n
)
d
x
a
x
2
+
b
x
+
c
=
{\displaystyle \int {\frac {(mx+n)\mathrm {d} x}{ax^{2}+bx+c}=}
m
2
a
ln
|
a
x
2
+
b
x
+
c
|
+
2
a
n
−
b
m
a
4
a
c
−
b
2
arctan
2
a
x
+
b
4
a
c
−
b
2
(pro
4
a
c
−
b
2
>
0
)
{\displaystyle {\frac {m}{2a}\ln \left|ax^{2}+bx+c\right|+{\frac {2an-bm}{a{\sqrt {4ac-b^{2}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}\qquad {\mbox{(pro }4ac-b^{2}>0{\mbox{)}
m
2
a
ln
|
a
x
2
+
b
x
+
c
|
+
2
a
n
−
b
m
a
b
2
−
4
a
c
a
r
t
a
n
h
2
a
x
+
b
b
2
−
4
a
c
(pro
4
a
c
−
b
2
<
0
)
{\displaystyle {\frac {m}{2a}\ln \left|ax^{2}+bx+c\right|+{\frac {2an-bm}{a{\sqrt {b^{2}-4ac}\,\mathrm {artanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}\qquad {\mbox{(pro }4ac-b^{2}<0{\mbox{)}
m
2
a
ln
|
a
x
2
+
b
x
+
c
|
−
2
a
n
−
b
m
a
(
2
a
x
+
b
)
(pro
4
a
c
−
b
2
=
0
)
{\displaystyle {\frac {m}{2a}\ln \left|ax^{2}+bx+c\right|-{\frac {2an-bm}{a(2ax+b)}\,\,\,\,\,\,\,\,\,\,\qquad {\mbox{(pro }4ac-b^{2}=0{\mbox{)}
∫
d
x
(
a
x
2
+
b
x
+
c
)
n
=
2
a
x
+
b
(
n
−
1
)
(
4
a
c
−
b
2
)
(
a
x
2
+
b
x
+
c
)
n
−
1
+
(
2
n
−
3
)
2
a
(
n
−
1
)
(
4
a
c
−
b
2
)
∫
d
x
(
a
x
2
+
b
x
+
c
)
n
−
1
{\displaystyle \int {\frac {\mathrm {d} x}{(ax^{2}+bx+c)^{n}={\frac {2ax+b}{(n-1)(4ac-b^{2})(ax^{2}+bx+c)^{n-1}+{\frac {(2n-3)2a}{(n-1)(4ac-b^{2})}\int {\frac {\mathrm {d} x}{(ax^{2}+bx+c)^{n-1}\,\!}
∫
x
d
x
(
a
x
2
+
b
x
+
c
)
n
=
b
x
+
2
c
(
n
−
1
)
(
4
a
c
−
b
2
)
(
a
x
2
+
b
x
+
c
)
n
−
1
−
b
(
2
n
−
3
)
(
n
−
1
)
(
4
a
c
−
b
2
)
∫
d
x
(
a
x
2
+
b
x
+
c
)
n
−
1
{\displaystyle \int {\frac {x\,\mathrm {d} x}{(ax^{2}+bx+c)^{n}={\frac {bx+2c}{(n-1)(4ac-b^{2})(ax^{2}+bx+c)^{n-1}-{\frac {b(2n-3)}{(n-1)(4ac-b^{2})}\int {\frac {\mathrm {d} x}{(ax^{2}+bx+c)^{n-1}\,\!}
∫
d
x
x
(
a
x
2
+
b
x
+
c
)
=
1
2
c
ln
|
x
2
a
x
2
+
b
x
+
c
|
−
b
2
c
∫
d
x
a
x
2
+
b
x
+
c
{\displaystyle \int {\frac {\mathrm {d} x}{x(ax^{2}+bx+c)}={\frac {1}{2c}\ln \left|{\frac {x^{2}{ax^{2}+bx+c}\right|-{\frac {b}{2c}\int {\frac {\mathrm {d} x}{ax^{2}+bx+c}
rozkladu na parciální zlomky , rozkladem racionální funkce na sumu výrazů ve tvaru:
e
x
+
f
(
a
x
2
+
b
x
+
c
)
n
{\displaystyle {\frac {ex+f}{\left(ax^{2}+bx+c\right)^{n}
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