Page:Annales de mathématiques pures et appliquées, 1829-1830, Tome 20.djvu/131

${\frac {\operatorname {d} ^{r}Q}{\operatorname {d} x^{r}}}=a^{x}Log.^{r}a\,;$

ce qui donnera, en substituant,

${\frac {\operatorname {d} ^{m}Pa^{x}}{\operatorname {d} x^{m}}}=a^{x}\operatorname {Log} .^{m}a\times$

$\left\{P+{\frac {m}{1}}{\frac {1}{\operatorname {Log} .a}}{\frac {\operatorname {d} P}{\operatorname {d} x}}+{\frac {m}{1}}.{\frac {m-1}{2}}{\frac {1}{\operatorname {Log} .^{2}a}}{\frac {\operatorname {d} ^{2}P}{\operatorname {d} x^{2}}}+\ldots \right\}\,;$

mais si l’on pose successivement

$P=\operatorname {Sin} .x,\qquad P=\operatorname {Cos} .x,$

on aura, dans le premier cas,

${\frac {\operatorname {d} P}{\operatorname {d} x}}=\operatorname {Cos} .x,\quad {\frac {\operatorname {d} ^{2}P}{\operatorname {d} x^{2}}}=-\operatorname {Sin} .x,\quad {\frac {\operatorname {d} ^{3}P}{\operatorname {d} x^{3}}}=-\operatorname {Cos} .x,\ldots ,$

et, dans le second,

${\frac {\operatorname {d} P}{\operatorname {d} x}}=-\operatorname {Sin} .x,\quad {\frac {\operatorname {d} ^{2}P}{\operatorname {d} x^{2}}}=-\operatorname {Cos} .x,\quad {\frac {\operatorname {d} ^{3}P}{\operatorname {d} x^{3}}}=+\operatorname {Sin} .x,\ldots ,$

substituant donc, tour à tour, dans la formule ci-dessus, elle deviendra, dans le premier cas,

${\frac {\operatorname {d} ^{m}.a^{x}\operatorname {Sin} .x}{\operatorname {d} x^{m}}}=a^{x}\operatorname {Log} .^{m}a\times$

$\left(\operatorname {Sin} .x+{\frac {m}{1}}{\frac {\operatorname {Cos} .x}{\operatorname {Log} .a}}-{\frac {m}{1}}.{\frac {m-1}{2}}{\frac {\operatorname {Sin} .x}{\operatorname {Log} .^{2}a}}-{\frac {m}{1}}.{\frac {m-1}{2}}{\frac {m-2}{3}}{\frac {\operatorname {Cos} .x}{\operatorname {Log} .^{3}a}}+\ldots \right)\,;$

et dans le second,

${\frac {\operatorname {d} ^{m}.a^{x}\operatorname {Cos} .x}{\operatorname {d} x^{m}}}=a^{x}\operatorname {Log} .^{m}a\times$

$\left(\operatorname {Cos} .x-{\frac {m}{1}}{\frac {\operatorname {Sin} .x}{\operatorname {Log} .a}}-{\frac {m}{1}}.{\frac {m-1}{2}}{\frac {\operatorname {Cos} .x}{\operatorname {Log} .^{2}a}}+{\frac {m}{1}}.{\frac {m-1}{2}}{\frac {m-2}{3}}{\frac {\operatorname {Sin} .x}{\operatorname {Log} .^{3}a}}+\ldots \right)\,;$

on aura donc en changeant le signe de $m$,

$\int ^{m}a^{x}\operatorname {Sin} .x\operatorname {d} x^{m}={\frac {a^{x}}{\operatorname {Log} .^{m}a}}\times$

$\left(\operatorname {Sin} .x-{\frac {m}{1}}{\frac {\operatorname {Cos} .x}{\operatorname {Log} .a}}-{\frac {m}{1}}.{\frac {m+1}{2}}{\frac {\operatorname {Sin} .x}{\operatorname {Log} .^{2}a}}+{\frac {m}{1}}.{\frac {m+1}{2}}{\frac {m+2}{3}}{\frac {\operatorname {Cos} .x}{\operatorname {Log} .^{3}a}}+\ldots \right),$