![{\displaystyle {\frac {\operatorname {d} ^{r}Q}{\operatorname {d} x^{r}}}=a^{x}Log.^{r}a\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2ae4a4e46e2f9047a2ae69621576ba8c8743b14)
ce qui donnera, en substituant,
![{\displaystyle {\frac {\operatorname {d} ^{m}Pa^{x}}{\operatorname {d} x^{m}}}=a^{x}\operatorname {Log} .^{m}a\times }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f598c3876e4240264db66af013bce983fdcca5b)
![{\displaystyle \left\{P+{\frac {m}{1}}{\frac {1}{\operatorname {Log} .a}}{\frac {\operatorname {d} P}{\operatorname {d} x}}+{\frac {m}{1}}.{\frac {m-1}{2}}{\frac {1}{\operatorname {Log} .^{2}a}}{\frac {\operatorname {d} ^{2}P}{\operatorname {d} x^{2}}}+\ldots \right\}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48bb80beabd8c42820a7128870abd8d891487bed)
mais si l’on pose successivement
![{\displaystyle P=\operatorname {Sin} .x,\qquad P=\operatorname {Cos} .x,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64a5923815a43e125132a39b8ae097fef90850ca)
on aura, dans le premier cas,
![{\displaystyle {\frac {\operatorname {d} P}{\operatorname {d} x}}=\operatorname {Cos} .x,\quad {\frac {\operatorname {d} ^{2}P}{\operatorname {d} x^{2}}}=-\operatorname {Sin} .x,\quad {\frac {\operatorname {d} ^{3}P}{\operatorname {d} x^{3}}}=-\operatorname {Cos} .x,\ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/264833ee0cd157d9ba932f06a41d8dcbeb8561ae)
et, dans le second,
![{\displaystyle {\frac {\operatorname {d} P}{\operatorname {d} x}}=-\operatorname {Sin} .x,\quad {\frac {\operatorname {d} ^{2}P}{\operatorname {d} x^{2}}}=-\operatorname {Cos} .x,\quad {\frac {\operatorname {d} ^{3}P}{\operatorname {d} x^{3}}}=+\operatorname {Sin} .x,\ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d417133e1362b76b7058f7364cbe6e8aca4ec3ef)
substituant donc, tour à tour, dans la formule ci-dessus, elle deviendra, dans le premier cas,
![{\displaystyle {\frac {\operatorname {d} ^{m}.a^{x}\operatorname {Sin} .x}{\operatorname {d} x^{m}}}=a^{x}\operatorname {Log} .^{m}a\times }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9658d73d8d1d851232171ed352567814d048d91)
![{\displaystyle \left(\operatorname {Sin} .x+{\frac {m}{1}}{\frac {\operatorname {Cos} .x}{\operatorname {Log} .a}}-{\frac {m}{1}}.{\frac {m-1}{2}}{\frac {\operatorname {Sin} .x}{\operatorname {Log} .^{2}a}}-{\frac {m}{1}}.{\frac {m-1}{2}}{\frac {m-2}{3}}{\frac {\operatorname {Cos} .x}{\operatorname {Log} .^{3}a}}+\ldots \right)\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91dcd5bd4309007efbcb761ac38e8fd574d9e7a5)
et dans le second,
![{\displaystyle {\frac {\operatorname {d} ^{m}.a^{x}\operatorname {Cos} .x}{\operatorname {d} x^{m}}}=a^{x}\operatorname {Log} .^{m}a\times }](https://wikimedia.org/api/rest_v1/media/math/render/svg/34a4d3520bee83df0d5637ce59462af276a6b631)
![{\displaystyle \left(\operatorname {Cos} .x-{\frac {m}{1}}{\frac {\operatorname {Sin} .x}{\operatorname {Log} .a}}-{\frac {m}{1}}.{\frac {m-1}{2}}{\frac {\operatorname {Cos} .x}{\operatorname {Log} .^{2}a}}+{\frac {m}{1}}.{\frac {m-1}{2}}{\frac {m-2}{3}}{\frac {\operatorname {Sin} .x}{\operatorname {Log} .^{3}a}}+\ldots \right)\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba0c92edd56b670f07de13720299f652247b814d)
on aura donc en changeant le signe de
,
![{\displaystyle \int ^{m}a^{x}\operatorname {Sin} .x\operatorname {d} x^{m}={\frac {a^{x}}{\operatorname {Log} .^{m}a}}\times }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b621d8192b506022d9d3decfa3c6ba6be95b6a2)
![{\displaystyle \left(\operatorname {Sin} .x-{\frac {m}{1}}{\frac {\operatorname {Cos} .x}{\operatorname {Log} .a}}-{\frac {m}{1}}.{\frac {m+1}{2}}{\frac {\operatorname {Sin} .x}{\operatorname {Log} .^{2}a}}+{\frac {m}{1}}.{\frac {m+1}{2}}{\frac {m+2}{3}}{\frac {\operatorname {Cos} .x}{\operatorname {Log} .^{3}a}}+\ldots \right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bba7a412998debda94e6c5a0b479a220ee0332c3)