394
QUESTIONS
![{\displaystyle \operatorname {f} (a)=\int {\frac {\operatorname {d} a}{1+a^{2}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e408a96800639332dad15698f376c8ab8d64e09a)
changeant donc, tour à tour,
en
et
on aura, au moyen de notre théorème général, pour la somme de la suite proposée,
![{\displaystyle {\frac {1}{2}}\left\{\int {\frac {a\operatorname {d} .\left(\operatorname {Cos} .x+{\sqrt {-1}}\operatorname {Sin} .x\right)}{1+a^{2}\left(\operatorname {Cos} .2x+{\sqrt {-1}}\operatorname {Sin} .2x\right)}}+\int {\frac {a\operatorname {d} .\left(\operatorname {Cos} .x-{\sqrt {-1}}\operatorname {Sin} .x\right)}{1+a^{2}\left(\operatorname {Cos} .2x-{\sqrt {-1}}\operatorname {Sin} .2x\right)}}\right\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d92f11c1e94db1829e4809eb37c6c75369555b0d)
ou, en exécutant les différentiations,
![{\displaystyle {\frac {a}{2}}\int \operatorname {d} x\left\{{\frac {-\operatorname {Sin} .x+{\sqrt {-1}}\operatorname {Cos} .x}{\left(1+a^{2}\operatorname {Cos} .2x\right)+{\sqrt {-1}}a^{2}\operatorname {Sin} .2x}}-{\frac {\operatorname {Sin} .x+{\sqrt {-1}}\operatorname {Cos} .x}{\left(1+a^{2}\operatorname {Cos} .2x\right)-{\sqrt {-1}}a^{2}\operatorname {Sin} .2x}}\right\},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0c160c691e07bc01e8d2f5be117d2ed0466d28c)
ou bien
![{\displaystyle {\frac {1}{2}}\int -{\frac {2\left(1-a^{2}\right)a\operatorname {d} x\operatorname {Sin} .x}{\left(1-a^{2}\right)+4a^{2}\operatorname {Cos} .^{2}x}}={\frac {1}{2}}\int -{\frac {\frac {2a\operatorname {d} x\operatorname {Sin} .x}{1-a^{2}}}{1+\left({\frac {2a\operatorname {Cos} .x}{1-a^{2}}}\right)^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9768fdaed2e35fba7ff03f349fdcd8cc2ad1aafe)
![{\displaystyle ={\frac {1}{2}}\operatorname {Arc} .\left(\operatorname {Tang} .={\frac {2a\operatorname {Cos} .x}{1-a^{2}}}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fe247bb10f55835e9786d719a87b03aa2e6178a)
comme nous l’avions déjà trouvé.
2. Pour sommer la suite
![{\displaystyle {\frac {\operatorname {Cos} .x}{1}}+{\frac {1}{2}}.{\frac {\operatorname {Cos} .3x}{3}}+{\frac {1.3}{2.4}}.{\frac {\operatorname {Cos} .5x}{5}}+{\frac {1.3.5}{2.4.6}}.{\frac {\operatorname {Cos} .7x}{7}}+\ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69f2bbabeabaa6514a8971659e96089fc6bf3ee9)
nous considérerons que
![{\displaystyle \int {\frac {\operatorname {d} a}{\sqrt {1-a^{2}}}}={\frac {a}{1}}+{\frac {1}{2}}.{\frac {a^{3}}{3}}+{\frac {1.3}{2.4}}.{\frac {a^{5}}{5}}+{\frac {1.3.5}{2.4.6}}.{\frac {a^{7}}{7}}+\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d3bdfce2467321f3982823e592d4cd7c47b5996)
d’où il suit que