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Question Number 153993 by ZiYangLee last updated on 12/Sep/21

Given that ∫_0 ^( 12) f(x) dx=20,  find the value of ∫_1 ^( 8)  ((f(4 log_2 x))/x) dx.

$$\mathrm{Given}\:\mathrm{that}\:\int_{\mathrm{0}} ^{\:\mathrm{12}} {f}\left({x}\right)\:{dx}=\mathrm{20}, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\int_{\mathrm{1}} ^{\:\mathrm{8}} \:\frac{{f}\left(\mathrm{4}\:\mathrm{log}_{\mathrm{2}} {x}\right)}{{x}}\:{dx}. \\ $$

Answered by mr W last updated on 12/Sep/21

let u=4log_2  x  x=2^(u/4)   dx=((ln 2)/4)×2^(u/4) du   ∫_1 ^( 8)  ((f(4 log_2 x))/x) dx=∫_0 ^(12) ((f(u))/2^(u/4) )×((ln 2)/4)×2^(u/4) du  =((ln 2)/4)∫_0 ^(12) f(u)du=5ln 2

$${let}\:{u}=\mathrm{4log}_{\mathrm{2}} \:{x} \\ $$$${x}=\mathrm{2}^{\frac{{u}}{\mathrm{4}}} \\ $$$${dx}=\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{4}}×\mathrm{2}^{\frac{{u}}{\mathrm{4}}} {du} \\ $$$$\:\int_{\mathrm{1}} ^{\:\mathrm{8}} \:\frac{{f}\left(\mathrm{4}\:\mathrm{log}_{\mathrm{2}} {x}\right)}{{x}}\:{dx}=\int_{\mathrm{0}} ^{\mathrm{12}} \frac{{f}\left({u}\right)}{\mathrm{2}^{\frac{{u}}{\mathrm{4}}} }×\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{4}}×\mathrm{2}^{\frac{{u}}{\mathrm{4}}} {du} \\ $$$$=\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{4}}\int_{\mathrm{0}} ^{\mathrm{12}} {f}\left({u}\right){du}=\mathrm{5ln}\:\mathrm{2} \\ $$

Commented by ZiYangLee last updated on 13/Sep/21

wow thanks mr W

$${wow}\:{thanks}\:{mr}\:{W} \\ $$

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