209
DÉFINIES.
différentes. En effet,
étant un nombre entier, on peut toujours lui donner la forme d’un multiple de
plus un nombre entier moindre que
en supposant ce dernier nombre également entier, ce qui donne les relations suivantes :
![{\displaystyle {\begin{array}{ll}\int u^{rn}\left(1-au^{n}\right)^{p}\operatorname {d} u&={\frac {1(1+n)\ldots \left[1+(r-1)n\right]}{a^{r}\left[1+(p+1)n\right]\ldots \left[1+(p+r)n\right]}}\int \left(1-au^{n}\right)^{p}\operatorname {d} u,\\\\\int u^{rn+1}\left(1-au^{n}\right)^{p}\operatorname {d} u&={\frac {2(2+n)\ldots \left[2+(r-1)n\right]}{a^{r}\left[2+(p+1)n\right]\ldots \left[2+(p+r)n\right]}}\int u\left(1-au^{n}\right)^{p}\operatorname {d} u,\\\ldots \ldots \ldots \ldots \ldots \ldots \ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\\\int u^{rn+n-1}\left(1-au^{n}\right)^{p}\operatorname {d} u&={\frac {n.2n.3n\ldots rn}{a^{r}(p+2)n\ldots (p+r+1)n}}\int u^{n-1}\left(1-au^{n}\right)^{p}\operatorname {d} u\,;\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7924ec2e213d7edd7e31385278cb9a8a28db5d2c)
d’où l’on déduit
![{\displaystyle {\begin{aligned}&\int \operatorname {d} u\left(1-au^{n}\right)^{p}\operatorname {F} (u)\\\\&=\left\{y_{0}+{\tfrac {1}{a\left[1+(p+1)n\right]}}y_{n}+{\tfrac {1(1+n)}{a^{2}\left[1+(p+1)n\right]\left[1+(p+2)n\right]}}y_{2n}+\ldots \right\}\int \left(1-au^{n}\right)^{p}\operatorname {d} u,\\\\&+\left\{y_{1}+{\tfrac {2}{a\left[2+(p+1)n\right]}}y_{n+1}+{\tfrac {2(2+n)}{a^{2}\left[2+(p+1)n\right]\left[2+(p+2)n\right]}}y_{2n+1}+\ldots \right\}\int \left(1-au^{n}\right)^{p}u\operatorname {d} u,\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&+\left\{y_{n-1}+{\tfrac {1}{a(p+2)}}y_{2n-1}+{\tfrac {1.2}{a^{2}(p+2)(p+3)}}y_{3n-1}+\ldots \right\}\int \left(1-au^{n}\right)^{p}u^{n-1}\operatorname {d} u\,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1808e041cbcd570c74fdb8e7fecb25640ba52450)
et, dans le cas particulier où
p
on aura
![{\displaystyle {\begin{aligned}\int \operatorname {d} u.e^{-bu^{n}}.\operatorname {F} (u)&=\left\{y_{0}+{\frac {1}{bn}}y_{n}+{\frac {1(1+n)}{b^{2}.n^{2}}}y_{2n}+\ldots \right\}\int e^{-bu^{n}}\operatorname {d} u\\\\&+\left\{y_{1}+{\frac {2}{bn}}y^{n+1}+{\frac {2(2+n)}{b^{2}.n^{2}}}y_{2n+1}+\ldots \right\}\int ue^{-bu^{n}}\operatorname {d} u\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&+\left\{y_{n-1}+{\frac {1}{b}}y^{2n+1}+{\frac {1.2}{b^{2}}}y_{3n-1}+\ldots \right\}\int u^{n-1}e^{-bu^{n}}\operatorname {d} u.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c689b09803f568bea8fb4a789edf6672dae5e329)