Accueil
Au hasard
Se connecter
Configuration
Faire un don
À propos de Wikisource
Avertissements
Rechercher
Page
:
Annales de mathématiques pures et appliquées, 1826-1827, Tome 17.djvu/128
Langue
Suivre
Modifier
Le texte de cette page a été
corrigé
et est conforme au fac-similé.
(174)
{
∫
0
ϖ
1
−
r
Cos
.
p
1
−
2
r
Cos
.
p
+
r
2
.
Cos
.
a
2
(
ϖ
−
p
)
(
Sin
.
p
2
)
a
d
p
=
2
a
−
1
.
ϖ
[
1
−
(
r
+
1
r
)
−
a
]
,
∫
0
ϖ
r
Sin
.
p
1
−
2
r
Cos
.
p
+
r
2
.
Sin
.
a
2
(
ϖ
−
p
)
(
Sin
.
p
2
)
a
d
p
=
2
a
−
1
.
ϖ
[
1
−
(
r
+
1
r
)
−
a
]
.
{\displaystyle \left\{{\begin{aligned}&\int _{0}^{\varpi }{\frac {1-r\operatorname {Cos} .p}{1-2r\operatorname {Cos} .p+r^{2}}}.{\frac {\operatorname {Cos} .{\frac {a}{2}}(\varpi -p)}{\left(\operatorname {Sin} .{\frac {p}{2}}\right)^{a}}}\operatorname {d} p\\&\qquad \qquad \qquad \qquad =2^{a-1}.\varpi \left[1-\left({\frac {r+1}{r}}\right)^{-a}\right],\\\\&\int _{0}^{\varpi }{\frac {r\operatorname {Sin} .p}{1-2r\operatorname {Cos} .p+r^{2}}}.{\frac {\operatorname {Sin} .{\frac {a}{2}}(\varpi -p)}{\left(\operatorname {Sin} .{\frac {p}{2}}\right)^{a}}}\operatorname {d} p\\&\qquad \qquad \qquad \qquad =2^{a-1}.\varpi \left[1-\left({\frac {r+1}{r}}\right)^{-a}\right].\end{aligned}}\right.}
(175)
{
∫
0
ϖ
1
−
r
Cos
.
p
1
−
2
r
Cos
.
p
+
r
2
.
Cos
.
a
2
(
ϖ
−
p
)
(
Sin
.
p
2
)
a
d
p
=
2
a
−
1
.
ϖ
[
1
−
(
r
−
1
r
)
−
a
]
,
∫
0
ϖ
r
Sin
.
p
1
−
2
r
Cos
.
p
+
r
2
.
Sin
.
a
2
(
ϖ
−
p
)
(
Sin
.
p
2
)
a
d
p
=
2
a
−
1
.
ϖ
[
1
−
(
r
−
1
r
)
−
a
]
.
{\displaystyle \left\{{\begin{aligned}&\int _{0}^{\varpi }{\frac {1-r\operatorname {Cos} .p}{1-2r\operatorname {Cos} .p+r^{2}}}.{\frac {\operatorname {Cos} .{\frac {a}{2}}(\varpi -p)}{\left(\operatorname {Sin} .{\frac {p}{2}}\right)^{a}}}\operatorname {d} p\\&\qquad \qquad \qquad \qquad =2^{a-1}.\varpi \left[1-\left({\frac {r-1}{r}}\right)^{-a}\right],\\\\&\int _{0}^{\varpi }{\frac {r\operatorname {Sin} .p}{1-2r\operatorname {Cos} .p+r^{2}}}.{\frac {\operatorname {Sin} .{\frac {a}{2}}(\varpi -p)}{\left(\operatorname {Sin} .{\frac {p}{2}}\right)^{a}}}\operatorname {d} p\\&\qquad \qquad \qquad \qquad =2^{a-1}.\varpi \left[1-\left({\frac {r-1}{r}}\right)^{-a}\right].\end{aligned}}\right.}
(176)
∫
0
ϖ
1
−
r
Cos
.
p
1
−
2
r
Cos
.
p
+
r
2
.
l
.
Tang
.
p
2
d
p
=
π
2
l
(
r
+
1
r
−
1
)
.
{\displaystyle \int _{0}^{\varpi }{\frac {1-r\operatorname {Cos} .p}{1-2r\operatorname {Cos} .p+r^{2}}}.\operatorname {l} .\operatorname {Tang} .{\frac {p}{2}}\operatorname {d} p={\frac {\pi }{2}}\operatorname {l} \left({\frac {r+1}{r-1}}\right).}
(177)
{
∫
0
ϖ
1
−
r
Cos
.
p
1
−
2
r
Cos
.
p
+
r
2
.
l
.
Cos
.
p
2
d
p
=
−
π
2
l
(
r
+
1
r
)
,
∫
0
ϖ
r
2
Sin
.
p
1
−
2
r
Cos
.
p
+
r
2
p
d
p
=
ϖ
l
(
r
+
1
r
)
.
{\displaystyle \left\{{\begin{aligned}&\int _{0}^{\varpi }{\frac {1-r\operatorname {Cos} .p}{1-2r\operatorname {Cos} .p+r^{2}}}.\operatorname {l} .\operatorname {Cos} .{\frac {p}{2}}\operatorname {d} p=-{\frac {\pi }{2}}\operatorname {l} \left({\frac {r+1}{r}}\right),\\\\&\int _{0}^{\varpi }{\frac {r^{2}\operatorname {Sin} .p}{1-2r\operatorname {Cos} .p+r^{2}}}p\operatorname {d} p=\varpi \operatorname {l} \left({\frac {r+1}{r}}\right).\end{aligned}}\right.}
(178)
∫
0
ϖ
1
−
r
Cos
.
p
1
−
2
r
Cos
.
p
+
r
2
.
l
.
Sin
.
p
2
d
p
=
π
2
l
(
r
r
−
1
)
.
{\displaystyle \int _{0}^{\varpi }{\frac {1-r\operatorname {Cos} .p}{1-2r\operatorname {Cos} .p+r^{2}}}.\operatorname {l} .\operatorname {Sin} .{\frac {p}{2}}\operatorname {d} p={\frac {\pi }{2}}\operatorname {l} \left({\frac {r}{r-1}}\right).}