Primitives de fonctions irrationnelles
Cet article dresse une liste non exhaustive de primitives de fonctions irrationnelles.
On suppose
a
≠
0
{\displaystyle a\neq 0}
.
∫
(
a
x
+
b
)
α
d
x
=
1
(
α
+
1
)
a
(
a
x
+
b
)
α
+
1
+
C
{\displaystyle \int (ax+b)^{\alpha }\,\mathrm {d} x={\frac {1}{(\alpha +1)a}}(ax+b)^{\alpha +1}+C}
(
α
≠
−
1
{\displaystyle \alpha \neq -1}
)
∫
1
a
x
2
+
b
x
+
c
d
x
{\displaystyle \int {\frac {1}{\sqrt {ax^{2}+bx+c}}}\,\mathrm {d} x}
=
{
1
a
arsinh
2
a
x
+
b
−
(
b
2
−
4
a
c
)
+
C
si
b
2
−
4
a
c
<
0
et
a
>
0
1
a
ln
|
2
a
x
+
b
|
+
C
si
b
2
−
4
a
c
=
0
et
a
>
0
−
1
−
a
arcsin
2
a
x
+
b
b
2
−
4
a
c
+
C
si
b
2
−
4
a
c
>
0
et
a
<
0
{\displaystyle ={\begin{cases}{\frac {1}{\sqrt {a}}}\operatorname {arsinh} {\frac {2ax+b}{\sqrt {-(b^{2}-4ac)}}}+C&{\text{si }}b^{2}-4ac<0{\text{ et }}a>0\\{\frac {1}{\sqrt {a}}}\ln |2ax+b|+C&{\text{si }}b^{2}-4ac=0{\text{ et }}a>0\\-{\frac {1}{\sqrt {-a}}}\operatorname {arcsin} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}+C&{\text{si }}b^{2}-4ac>0{\text{ et }}a<0\\\end{cases}}}
∫
a
x
2
+
b
x
+
c
d
x
=
2
a
x
+
b
4
a
a
x
2
+
b
x
+
c
−
b
2
−
4
a
c
8
a
∫
1
a
x
2
+
b
x
+
c
d
x
{\displaystyle \int {\sqrt {ax^{2}+bx+c}}\,\mathrm {d} x={\frac {2ax+b}{4a}}{\sqrt {ax^{2}+bx+c}}-{\frac {b^{2}-4ac}{8a}}\int {\frac {1}{\sqrt {ax^{2}+bx+c}}}\,\mathrm {d} x}
∫
x
a
x
2
+
b
x
+
c
d
x
=
a
x
2
+
b
x
+
c
a
−
b
2
a
∫
1
a
x
2
+
b
x
+
c
d
x
{\displaystyle \int {\frac {x}{\sqrt {ax^{2}+bx+c}}}\,\mathrm {d} x={\frac {\sqrt {ax^{2}+bx+c}}{a}}-{\frac {b}{2a}}\int {\frac {1}{\sqrt {ax^{2}+bx+c}}}\,\mathrm {d} x}
On suppose
a
>
0
{\displaystyle a>0}
∫
1
a
2
−
x
2
d
x
=
arcsin
x
a
+
C
{\displaystyle \int {\frac {1}{\sqrt {a^{2}-x^{2}}}}\,\mathrm {d} x=\operatorname {arcsin} {\frac {x}{a}}+C}
∫
1
a
2
+
x
2
d
x
=
arsinh
x
a
+
C
{\displaystyle \int {\frac {1}{\sqrt {a^{2}+x^{2}}}}\,\mathrm {d} x=\operatorname {arsinh} {\frac {x}{a}}+C}
∫
1
x
2
−
a
2
d
x
=
arcosh
x
a
+
C
{\displaystyle \int {\frac {1}{\sqrt {x^{2}-a^{2}}}}\,\mathrm {d} x=\operatorname {arcosh} {\frac {x}{a}}+C}
∫
a
2
−
x
2
d
x
=
x
2
a
2
−
x
2
+
a
2
2
arcsin
x
a
+
C
{\displaystyle \int {\sqrt {a^{2}-x^{2}}}\,\mathrm {d} x={\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}+{\frac {a^{2}}{2}}\operatorname {arcsin} {\frac {x}{a}}+C}
∫
a
2
+
x
2
d
x
=
x
2
a
2
+
x
2
+
a
2
2
arsinh
x
a
+
C
{\displaystyle \int {\sqrt {a^{2}+x^{2}}}\,\mathrm {d} x={\frac {x}{2}}{\sqrt {a^{2}+x^{2}}}+{\frac {a^{2}}{2}}\operatorname {arsinh} {\frac {x}{a}}+C}
∫
x
2
−
a
2
d
x
=
x
2
x
2
−
a
2
−
a
2
2
arcosh
x
a
+
C
{\displaystyle \int {\sqrt {x^{2}-a^{2}}}\,\mathrm {d} x={\frac {x}{2}}{\sqrt {x^{2}-a^{2}}}-{\frac {a^{2}}{2}}\operatorname {arcosh} {\frac {x}{a}}+C}
∫
x
a
2
+
x
2
d
x
=
1
3
(
a
2
+
x
2
)
3
+
C
{\displaystyle \int x{\sqrt {a^{2}+x^{2}}}\,\mathrm {d} x={\frac {1}{3}}{\sqrt {(a^{2}+x^{2})^{3}}}+C}
∫
x
a
2
−
x
2
d
x
=
−
1
3
(
a
2
−
x
2
)
3
+
C
{\displaystyle \int x{\sqrt {a^{2}-x^{2}}}\,\mathrm {d} x=-{\frac {1}{3}}{\sqrt {(a^{2}-x^{2})^{3}}}+C}
∫
x
x
2
−
a
2
d
x
=
1
3
(
x
2
−
a
2
)
3
+
C
{\displaystyle \int x{\sqrt {x^{2}-a^{2}}}\,\mathrm {d} x={\frac {1}{3}}{\sqrt {(x^{2}-a^{2})^{3}}}+C}
∫
1
x
a
2
+
x
2
d
x
=
a
2
+
x
2
−
a
ln
|
1
x
(
a
+
a
2
+
x
2
)
|
+
C
{\displaystyle \int {\frac {1}{x}}{\sqrt {a^{2}+x^{2}}}\,\mathrm {d} x={\sqrt {a^{2}+x^{2}}}-a\ln \left|{\frac {1}{x}}\left(a+{\sqrt {a^{2}+x^{2}}}\right)\right|+C}
∫
1
x
a
2
−
x
2
d
x
=
a
2
−
x
2
−
a
ln
|
1
x
(
a
+
a
2
−
x
2
)
|
+
C
{\displaystyle \int {\frac {1}{x}}{\sqrt {a^{2}-x^{2}}}\,\mathrm {d} x={\sqrt {a^{2}-x^{2}}}-a\ln \left|{\frac {1}{x}}\left(a+{\sqrt {a^{2}-x^{2}}}\right)\right|+C}
∫
1
x
x
2
−
a
2
d
x
=
x
2
−
a
2
−
a
arccos
a
x
+
C
{\displaystyle \int {\frac {1}{x}}{\sqrt {x^{2}-a^{2}}}\,\mathrm {d} x={\sqrt {x^{2}-a^{2}}}-a\operatorname {arccos} {\frac {a}{x}}+C}
∫
x
a
2
−
x
2
d
x
=
−
a
2
−
x
2
+
C
{\displaystyle \int {\frac {x}{\sqrt {a^{2}-x^{2}}}}\,\mathrm {d} x=-{\sqrt {a^{2}-x^{2}}}+C}
∫
x
a
2
+
x
2
d
x
=
a
2
+
x
2
+
C
{\displaystyle \int {\frac {x}{\sqrt {a^{2}+x^{2}}}}\,\mathrm {d} x={\sqrt {a^{2}+x^{2}}}+C}
∫
x
x
2
−
a
2
d
x
=
x
2
−
a
2
+
C
{\displaystyle \int {\frac {x}{\sqrt {x^{2}-a^{2}}}}\,\mathrm {d} x={\sqrt {x^{2}-a^{2}}}+C}
∫
x
2
a
2
−
x
2
d
x
=
−
x
2
a
2
−
x
2
+
a
2
2
arcsin
x
a
+
C
{\displaystyle \int {\frac {x^{2}}{\sqrt {a^{2}-x^{2}}}}\,\mathrm {d} x=-{\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}+{\frac {a^{2}}{2}}\operatorname {arcsin} {\frac {x}{a}}+C}
∫
x
2
a
2
+
x
2
d
x
=
x
2
a
2
+
x
2
−
a
2
2
arsinh
x
a
+
C
{\displaystyle \int {\frac {x^{2}}{\sqrt {a^{2}+x^{2}}}}\,\mathrm {d} x={\frac {x}{2}}{\sqrt {a^{2}+x^{2}}}-{\frac {a^{2}}{2}}\operatorname {arsinh} {\frac {x}{a}}+C}
∫
x
2
x
2
−
a
2
d
x
=
x
2
x
2
−
a
2
+
a
2
2
arcosh
x
a
+
C
{\displaystyle \int {\frac {x^{2}}{\sqrt {x^{2}-a^{2}}}}\,\mathrm {d} x={\frac {x}{2}}{\sqrt {x^{2}-a^{2}}}+{\frac {a^{2}}{2}}\operatorname {arcosh} {\frac {x}{a}}+C}
Portail de l'analyse
Synopsis CSI : X2
La suite de la version X de la célèbre série CSI..