Page:Annales de mathématiques pures et appliquées, 1826-1827, Tome 17.djvu/128

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

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

(176)$\int _{0}^{\varpi }{\frac {1-r\operatorname {Cos} .p}{1-2r\operatorname {Cos} .p+r^{2}}}.\operatorname {l} .\operatorname {Tang} .{\frac {p}{2}}\operatorname {d} p={\frac {\pi }{2}}\operatorname {l} \left({\frac {r+1}{r-1}}\right).$

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

(178)$\int _{0}^{\varpi }{\frac {1-r\operatorname {Cos} .p}{1-2r\operatorname {Cos} .p+r^{2}}}.\operatorname {l} .\operatorname {Sin} .{\frac {p}{2}}\operatorname {d} p={\frac {\pi }{2}}\operatorname {l} \left({\frac {r}{r-1}}\right).$