Page:Annales de mathématiques pures et appliquées, 1826-1827, Tome 17.djvu/126

(164)$\left\{{\begin{aligned}&\int _{0}^{\varpi }{\frac {\operatorname {Sin} .p}{r-\operatorname {Cos} .p}}p\operatorname {d} p=+\varpi \operatorname {l} (2+2r),\\\\&\int _{0}^{\varpi }{\frac {\operatorname {Sin} .p}{r+\operatorname {Cos} .p}}p\operatorname {d} p=-\varpi \operatorname {l} (2+2r)\,;\end{aligned}}\right.$

et pour r<4,

(165)$\left\{{\begin{aligned}&\int _{0}^{\varpi }{\frac {\operatorname {Sin} .p}{r-\operatorname {Cos} .p}}p\operatorname {d} p=+\varpi \operatorname {l} (2r+2)-\varpi \operatorname {l} \left(r+{\sqrt {r^{2}-1}}\right),\\\\&\int _{0}^{\varpi }{\frac {\operatorname {Sin} .p}{r+\operatorname {Cos} .p}}p\operatorname {d} p=-\varpi \operatorname {l} (2r-2)+\varpi \operatorname {l} \left(r+{\sqrt {r^{2}-1}}\right).\end{aligned}}\right.$

On trouvera encore, pour r<1,

(166)$\left\{{\begin{aligned}&\int _{0}^{\varpi }{\frac {1-r\operatorname {Cos} .p}{1-2r\operatorname {Cos} .p+r^{2}}}\left(\operatorname {Tang} .{\frac {p}{2}}\right)^{a}\operatorname {d} p={\frac {\pi }{2\operatorname {Cos} .{\frac {a\pi }{2}}}}\left[1-\left({\frac {1-r}{1+r}}\right)^{a}\right],\\\\&\int _{0}^{\varpi }{\frac {r\operatorname {Sin} .p}{1-2r\operatorname {Cos} .p+r^{2}}}\left(\operatorname {Tang} .{\frac {p}{2}}\right)^{a}\operatorname {d} p={\frac {\pi }{2\operatorname {Sin} .{\frac {a\pi }{2}}}}\left[1-\left({\frac {1-r}{1+r}}\right)^{a}\right].\end{aligned}}\right.$

(167)$\left\{{\begin{aligned}&\int _{0}^{\varpi }{\frac {1-r\operatorname {Cos} .p}{1-2r\operatorname {Cos} .p+r^{2}}}.{\frac {\operatorname {Cos} .{\frac {ap}{2}}}{\left(\operatorname {Cos} .{\frac {p}{2}}\right)^{a}}}.\operatorname {d} p=2^{a-1}.\varpi \left[1+(1+r)^{-a}\right],\\\\&\int _{0}^{\varpi }{\frac {r\operatorname {Sin} .p}{1-2r\operatorname {Cos} .p+r^{2}}}.{\frac {\operatorname {Sin} .{\frac {ap}{2}}}{\left(\operatorname {Cos} .{\frac {p}{2}}\right)^{a}}}.\operatorname {d} p=2^{a-1}.\varpi \left[1-(1+r)^{-a}\right].\end{aligned}}\right.$