Uit Wikipedia, de vrije encyclopedie
Naar navigatie springen
Naar zoeken springen
Hieronder staat een lijst van integralen van rationale functies . Integralen zijn het onderwerp van studie van de integraalrekening . Een rationale functie is een breuk waarvan zowel de teller als de noemer een polynoom is of gelijk is aan 1.
∫
(
a
x
+
b
)
n
d
x
=
(
a
x
+
b
)
n
+
1
a
(
n
+
1
)
voor
n
≠
−
1
{\displaystyle \int (ax+b)^{n}\ \mathrm {d} x={\frac {(ax+b)^{n+1}}{a(n+1)}}\qquad {\mbox{voor }}n\neq -1}
∫
1
a
x
+
b
d
x
=
1
a
ln
|
a
x
+
b
|
{\displaystyle \int {\frac {1}{ax+b}}\ \mathrm {d} x={\frac {1}{a}}\ln |ax+b|}
∫
x
(
a
x
+
b
)
n
d
x
=
a
(
n
+
1
)
x
−
b
a
2
(
n
+
1
)
(
n
+
2
)
(
a
x
+
b
)
n
+
1
voor
n
∉
{
−
1
,
−
2
}
{\displaystyle \int x(ax+b)^{n}\ \mathrm {d} x={\frac {a(n+1)x-b}{a^{2}(n+1)(n+2)}}(ax+b)^{n+1}\qquad {\mbox{voor }}n\not \in \{-1,-2\}}
∫
x
a
x
+
b
d
x
=
x
a
−
b
a
2
ln
|
a
x
+
b
|
{\displaystyle \int {\frac {x}{ax+b}}\ \mathrm {d} x={\frac {x}{a}}-{\frac {b}{a^{2}}}\ln |ax+b|}
∫
x
(
a
x
+
b
)
2
d
x
=
b
a
2
(
a
x
+
b
)
+
1
a
2
ln
|
a
x
+
b
|
{\displaystyle \int {\frac {x}{(ax+b)^{2}}}\ \mathrm {d} x={\frac {b}{a^{2}(ax+b)}}+{\frac {1}{a^{2}}}\ln |ax+b|}
∫
x
(
a
x
+
b
)
n
d
x
=
a
(
1
−
n
)
x
−
b
a
2
(
n
−
1
)
(
n
−
2
)
(
a
x
+
b
)
n
−
1
voor
n
∉
{
1
,
2
}
{\displaystyle \int {\frac {x}{(ax+b)^{n}}}\ \mathrm {d} x={\frac {a(1-n)x-b}{a^{2}(n-1)(n-2)(ax+b)^{n-1}}}\qquad {\mbox{voor }}n\not \in \{1,2\}}
∫
x
2
a
x
+
b
d
x
=
b
2
ln
|
a
x
+
b
|
a
3
+
a
x
2
−
2
b
x
2
a
2
{\displaystyle \int {\frac {x^{2}}{ax+b}}\ \mathrm {d} x={\frac {b^{2}\ln |ax+b|}{a^{3}}}+{\frac {ax^{2}-2bx}{2a^{2}}}}
∫
x
2
(
a
x
+
b
)
2
d
x
=
1
a
3
(
a
x
−
2
b
ln
|
a
x
+
b
|
−
b
2
a
x
+
b
)
{\displaystyle \int {\frac {x^{2}}{(ax+b)^{2}}}\ \mathrm {d} x={\frac {1}{a^{3}}}\left(ax-2b\ln |ax+b|-{\frac {b^{2}}{ax+b}}\right)}
∫
x
2
(
a
x
+
b
)
3
d
x
=
1
a
3
(
ln
|
a
x
+
b
|
+
2
b
a
x
+
b
−
b
2
2
(
a
x
+
b
)
2
)
{\displaystyle \int {\frac {x^{2}}{(ax+b)^{3}}}\ \mathrm {d} x={\frac {1}{a^{3}}}\left(\ln |ax+b|+{\frac {2b}{ax+b}}-{\frac {b^{2}}{2(ax+b)^{2}}}\right)}
∫
x
2
(
a
x
+
b
)
n
d
x
=
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
)
)
voor
n
∉
{
1
,
2
,
3
}
{\displaystyle \int {\frac {x^{2}}{(ax+b)^{n}}}\ \mathrm {d} x={\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{voor }}n\not \in \{1,2,3\}}
∫
1
x
(
a
x
+
b
)
d
x
=
−
1
b
ln
|
a
x
+
b
x
|
{\displaystyle \int {\frac {1}{x(ax+b)}}\ \mathrm {d} x=-{\frac {1}{b}}\ln \left|{\frac {ax+b}{x}}\right|}
∫
1
x
2
(
a
x
+
b
)
d
x
=
−
1
b
x
+
a
b
2
ln
|
a
x
+
b
x
|
{\displaystyle \int {\frac {1}{x^{2}(ax+b)}}\ \mathrm {d} x=-{\frac {1}{bx}}+{\frac {a}{b^{2}}}\ln \left|{\frac {ax+b}{x}}\right|}
∫
1
x
2
(
a
x
+
b
)
2
d
x
=
−
a
(
1
b
2
(
a
x
+
b
)
+
1
a
b
2
x
−
2
b
3
ln
|
a
x
+
b
x
|
)
{\displaystyle \int {\frac {1}{x^{2}(ax+b)^{2}}}\ \mathrm {d} x=-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)}
∫
1
x
2
+
a
2
d
x
=
1
a
arctan
x
a
{\displaystyle \int {\frac {1}{x^{2}+a^{2}}}\ \mathrm {d} x={\frac {1}{a}}\arctan {\frac {x}{a}}}
∫
1
x
2
−
a
2
d
x
=
{
−
1
a
a
r
c
t
a
n
h
x
a
=
1
2
a
ln
a
−
x
a
+
x
voor
|
x
|
<
|
a
|
−
1
a
a
r
c
c
o
t
h
x
a
=
1
2
a
ln
x
−
a
x
+
a
voor
|
x
|
>
|
a
|
{\displaystyle \int {\frac {1}{x^{2}-a^{2}}}\ \mathrm {d} x={\begin{cases}-{\frac {1}{a}}\ \mathrm {arctanh} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {a-x}{a+x}}&{\mbox{voor }}|x|<|a|\\-{\frac {1}{a}}\ \mathrm {arccoth} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {x-a}{x+a}}&{\mbox{voor }}|x|>|a|\end{cases}}}
∫
d
x
x
2
n
+
1
=
∑
k
=
1
2
n
−
1
{
1
2
n
−
1
[
sin
(
(
2
k
−
1
)
π
2
n
)
arctan
[
(
x
−
cos
(
(
2
k
−
1
)
π
2
n
)
)
csc
(
(
2
k
−
1
)
π
2
n
)
]
]
−
1
2
n
[
cos
(
(
2
k
−
1
)
π
2
n
)
ln
|
x
2
−
2
x
cos
(
(
2
k
−
1
)
π
2
n
)
+
1
|
]
}
{\displaystyle \int {\frac {\ \mathrm {d} x}{x^{2^{n}}+1}}=\sum _{k=1}^{2^{n-1}}\left\{{\frac {1}{2^{n-1}}}\left[\sin({\frac {(2k-1)\pi }{2^{n}}})\arctan[\left(x-\cos({\frac {(2k-1)\pi }{2^{n}}})\right)\csc({\frac {(2k-1)\pi }{2^{n}}})]\right]-{\frac {1}{2^{n}}}\left[\cos({\frac {(2k-1)\pi }{2^{n}}})\ln \left|x^{2}-2x\cos({\frac {(2k-1)\pi }{2^{n}}})+1\right|\right]\right\}}
voor a ≠ 0
∫
1
a
x
2
+
b
x
+
c
d
x
=
{
2
4
a
c
−
b
2
arctan
2
a
x
+
b
4
a
c
−
b
2
voor
4
a
c
−
b
2
>
0
−
2
b
2
−
4
a
c
a
r
c
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
|
voor
4
a
c
−
b
2
<
0
−
2
2
a
x
+
b
voor
4
a
c
−
b
2
=
0
{\displaystyle \int {\frac {1}{ax^{2}+bx+c}}\ \mathrm {d} x={\begin{cases}{\frac {2}{\sqrt {4ac-b^{2}}}}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}&{\mbox{voor }}4ac-b^{2}>0\\-{\frac {2}{\sqrt {b^{2}-4ac}}}\ \mathrm {arctanh} {\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|&{\mbox{voor }}4ac-b^{2}<0\\-{\frac {2}{2ax+b}}&{\mbox{voor }}4ac-b^{2}=0\end{cases}}}
∫
x
a
x
2
+
b
x
+
c
d
x
{\displaystyle \int {\frac {x}{ax^{2}+bx+c}}\ \mathrm {d} x}
=
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
a
x
2
+
b
x
+
c
d
x
=
{
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
voor
4
a
c
−
b
2
>
0
m
2
a
ln
|
a
x
2
+
b
x
+
c
‖
−
2
a
n
−
b
m
a
b
2
−
4
a
c
a
r
c
t
a
n
h
2
a
x
+
b
b
2
−
4
a
c
voor
4
a
c
−
b
2
<
0
m
2
a
ln
|
a
x
2
+
b
x
+
c
|
−
2
a
n
−
b
m
a
(
2
a
x
+
b
)
voor
4
a
c
−
b
2
=
0
{\displaystyle \int {\frac {mx+n}{ax^{2}+bx+c}}\ \mathrm {d} x={\begin{cases}{\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}}}}&{\mbox{voor }}4ac-b^{2}>0\\{\frac {m}{2a}}\ln |ax^{2}+bx+c\|-{\frac {2an-bm}{a{\sqrt {b^{2}-4ac}}}}\ \mathrm {arctanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}&{\mbox{voor }}4ac-b^{2}<0\\{\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|-{\frac {2an-bm}{a(2ax+b)}}&{\mbox{voor }}4ac-b^{2}=0\end{cases}}}
∫
1
(
a
x
2
+
b
x
+
c
)
n
d
x
=
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
)
∫
1
(
a
x
2
+
b
x
+
c
)
n
−
1
d
x
{\displaystyle \int {\frac {1}{(ax^{2}+bx+c)^{n}}}\ \mathrm {d} x={\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 {1}{(ax^{2}+bx+c)^{n-1}}}\ \mathrm {d} x}
∫
x
(
a
x
2
+
b
x
+
c
)
n
d
x
=
−
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
)
∫
1
(
a
x
2
+
b
x
+
c
)
n
−
1
d
x
{\displaystyle \int {\frac {x}{(ax^{2}+bx+c)^{n}}}\ \mathrm {d} x=-{\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 {1}{(ax^{2}+bx+c)^{n-1}}}\ \mathrm {d} x}
∫
1
x
(
a
x
2
+
b
x
+
c
)
d
x
=
1
2
c
ln
|
x
2
a
x
2
+
b
x
+
c
|
−
b
2
c
∫
1
a
x
2
+
b
x
+
c
d
x
{\displaystyle \int {\frac {1}{x(ax^{2}+bx+c)}}\ \mathrm {d} x={\frac {1}{2c}}\ln \left|{\frac {x^{2}}{ax^{2}+bx+c}}\right|-{\frac {b}{2c}}\int {\frac {1}{ax^{2}+bx+c}}\ \mathrm {d} x}
met een wortel in de noemer
∫
a
x
n
b
c
x
m
d
x
=
a
x
n
+
1
b
(
−
m
2
+
n
+
1
)
c
x
m
{\displaystyle \int {\frac {ax^{n}}{b{\sqrt {cx^{m}}}}}\ \mathrm {d} x={\frac {ax^{n+1}}{b\left(-{\frac {m}{2}}+n+1\right){\sqrt {cx^{m}}}}}}
![{\displaystyle \int {\frac {\ \mathrm {d} x}{x^{2^{n}}+1}}=\sum _{k=1}^{2^{n-1}}\left\{{\frac {1}{2^{n-1}}}\left[\sin({\frac {(2k-1)\pi }{2^{n}}})\arctan[\left(x-\cos({\frac {(2k-1)\pi }{2^{n}}})\right)\csc({\frac {(2k-1)\pi }{2^{n}}})]\right]-{\frac {1}{2^{n}}}\left[\cos({\frac {(2k-1)\pi }{2^{n}}})\ln \left|x^{2}-2x\cos({\frac {(2k-1)\pi }{2^{n}}})+1\right|\right]\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf3c217bdfed193d6b0a8ad551c33f2f8fb3580)
