Question Number 153993 by ZiYangLee last updated on 12/Sep/21
$$\mathrm{Given}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\:\mathrm{12}} {f}\left({x}\right)\:{dx}=\mathrm{20}, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\int_{\mathrm{1}} ^{\:\mathrm{8}} \:\frac{{f}\left(\mathrm{4}\:\mathrm{log}_{\mathrm{2}} {x}\right)}{{x}}\:{dx}. \\ $$
Answered by mr W last updated on 12/Sep/21
$${let}\:{u}=\mathrm{4log}_{\mathrm{2}} \:{x} \\ $$$${x}=\mathrm{2}^{\frac{{u}}{\mathrm{4}}} \\ $$$${dx}=\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{4}}×\mathrm{2}^{\frac{{u}}{\mathrm{4}}} {du} \\ $$$$\:\int_{\mathrm{1}} ^{\:\mathrm{8}} \:\frac{{f}\left(\mathrm{4}\:\mathrm{log}_{\mathrm{2}} {x}\right)}{{x}}\:{dx}=\int_{\mathrm{0}} ^{\mathrm{12}} \frac{{f}\left({u}\right)}{\mathrm{2}^{\frac{{u}}{\mathrm{4}}} }×\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{4}}×\mathrm{2}^{\frac{{u}}{\mathrm{4}}} {du} \\ $$$$=\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{4}}\int_{\mathrm{0}} ^{\mathrm{12}} {f}\left({u}\right){du}=\mathrm{5ln}\:\mathrm{2} \\ $$
Commented by ZiYangLee last updated on 13/Sep/21
$${wow}\:{thanks}\:{mr}\:{W} \\ $$