Question Number 41851 by de best last updated on 13/Aug/18
Commented by math khazana by abdo last updated on 14/Aug/18
$${changement}\:{y}={atan}\left({t}\right)\:{give} \\ $$$$\int\:\:\:\frac{{dy}}{{a}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }\:=\frac{\mathrm{1}}{{a}^{\mathrm{2}} }\:\:\int\:\:\:\:\frac{{a}\left(\mathrm{1}+{tan}^{\mathrm{2}} {t}\right){dt}}{\mathrm{1}+{tan}^{\mathrm{2}} {t}}\:\:\:\:\left({a}\neq\mathrm{0}\right) \\ $$$$=\:\frac{\mathrm{1}}{{a}}{t}\:+{c}\:=\frac{\mathrm{1}}{{a}}\:{arctan}\left(\frac{{y}}{{a}}\right)+{c} \\ $$$${if}\:{a}=\mathrm{0}\:\:{we}\:{get}?\:\int\:\:\:\frac{{dy}}{{y}^{\mathrm{2}} }\:=−\frac{\mathrm{1}}{{y}}\:+{c}\:\:. \\ $$
Commented by de best last updated on 14/Aug/18
$${thanks}\:{sir}.\:{really}\:{appreciate} \\ $$