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Annales de mathématiques pures et appliquées, 1826-1827, Tome 17.djvu/126
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(164)
{
∫
0
ϖ
Sin
.
p
r
−
Cos
.
p
p
d
p
=
+
ϖ
l
(
2
+
2
r
)
,
∫
0
ϖ
Sin
.
p
r
+
Cos
.
p
p
d
p
=
−
ϖ
l
(
2
+
2
r
)
;
{\displaystyle \left\{{\begin{aligned}&\int _{0}^{\varpi }{\frac {\operatorname {Sin} .p}{r-\operatorname {Cos} .p}}p\operatorname {d} p=+\varpi \operatorname {l} (2+2r),\\\\&\int _{0}^{\varpi }{\frac {\operatorname {Sin} .p}{r+\operatorname {Cos} .p}}p\operatorname {d} p=-\varpi \operatorname {l} (2+2r)\,;\end{aligned}}\right.}
et pour r<4,
(165)
{
∫
0
ϖ
Sin
.
p
r
−
Cos
.
p
p
d
p
=
+
ϖ
l
(
2
r
+
2
)
−
ϖ
l
(
r
+
r
2
−
1
)
,
∫
0
ϖ
Sin
.
p
r
+
Cos
.
p
p
d
p
=
−
ϖ
l
(
2
r
−
2
)
+
ϖ
l
(
r
+
r
2
−
1
)
.
{\displaystyle \left\{{\begin{aligned}&\int _{0}^{\varpi }{\frac {\operatorname {Sin} .p}{r-\operatorname {Cos} .p}}p\operatorname {d} p=+\varpi \operatorname {l} (2r+2)-\varpi \operatorname {l} \left(r+{\sqrt {r^{2}-1}}\right),\\\\&\int _{0}^{\varpi }{\frac {\operatorname {Sin} .p}{r+\operatorname {Cos} .p}}p\operatorname {d} p=-\varpi \operatorname {l} (2r-2)+\varpi \operatorname {l} \left(r+{\sqrt {r^{2}-1}}\right).\end{aligned}}\right.}
On trouvera encore, pour r<1,
(166)
{
∫
0
ϖ
1
−
r
Cos
.
p
1
−
2
r
Cos
.
p
+
r
2
(
Tang
.
p
2
)
a
d
p
=
π
2
Cos
.
a
π
2
[
1
−
(
1
−
r
1
+
r
)
a
]
,
∫
0
ϖ
r
Sin
.
p
1
−
2
r
Cos
.
p
+
r
2
(
Tang
.
p
2
)
a
d
p
=
π
2
Sin
.
a
π
2
[
1
−
(
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}}}\left(\operatorname {Tang} .{\frac {p}{2}}\right)^{a}\operatorname {d} p={\frac {\pi }{2\operatorname {Cos} .{\frac {a\pi }{2}}}}\left[1-\left({\frac {1-r}{1+r}}\right)^{a}\right],\\\\&\int _{0}^{\varpi }{\frac {r\operatorname {Sin} .p}{1-2r\operatorname {Cos} .p+r^{2}}}\left(\operatorname {Tang} .{\frac {p}{2}}\right)^{a}\operatorname {d} p={\frac {\pi }{2\operatorname {Sin} .{\frac {a\pi }{2}}}}\left[1-\left({\frac {1-r}{1+r}}\right)^{a}\right].\end{aligned}}\right.}
(167)
{
∫
0
ϖ
1
−
r
Cos
.
p
1
−
2
r
Cos
.
p
+
r
2
.
Cos
.
a
p
2
(
Cos
.
p
2
)
a
.
d
p
=
2
a
−
1
.
ϖ
[
1
+
(
1
+
r
)
−
a
]
,
∫
0
ϖ
r
Sin
.
p
1
−
2
r
Cos
.
p
+
r
2
.
Sin
.
a
p
2
(
Cos
.
p
2
)
a
.
d
p
=
2
a
−
1
.
ϖ
[
1
−
(
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 {ap}{2}}}{\left(\operatorname {Cos} .{\frac {p}{2}}\right)^{a}}}.\operatorname {d} p=2^{a-1}.\varpi \left[1+(1+r)^{-a}\right],\\\\&\int _{0}^{\varpi }{\frac {r\operatorname {Sin} .p}{1-2r\operatorname {Cos} .p+r^{2}}}.{\frac {\operatorname {Sin} .{\frac {ap}{2}}}{\left(\operatorname {Cos} .{\frac {p}{2}}\right)^{a}}}.\operatorname {d} p=2^{a-1}.\varpi \left[1-(1+r)^{-a}\right].\end{aligned}}\right.}