Question-185708

Question Number 185708 by mokys last updated on 26/Jan/23

Answered by mr W last updated on 26/Jan/23

t=4 log_2 x=((4 ln x)/(ln 2)) dt=(4/(x ln 2))dx ∫_1 ^8 ((f(4 log_2 x))/x)dx =((ln 2)/4)∫_0 ^(12) f(t)dt =((ln 2)/4)×20 =5 ln 2

$${t}=\mathrm{4}\:\mathrm{log}_{\mathrm{2}} \:{x}=\frac{\mathrm{4}\:\mathrm{ln}\:{x}}{\mathrm{ln}\:\mathrm{2}} \\ $$$${dt}=\frac{\mathrm{4}}{{x}\:\mathrm{ln}\:\mathrm{2}}{dx} \\ $$$$\int_{\mathrm{1}} ^{\mathrm{8}} \frac{{f}\left(\mathrm{4}\:\mathrm{log}_{\mathrm{2}} \:{x}\right)}{{x}}{dx} \\ $$$$=\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{4}}\int_{\mathrm{0}} ^{\mathrm{12}} {f}\left({t}\right){dt} \\ $$$$=\frac{\mathrm{ln}\:\mathrm{2}}{\mathrm{4}}×\mathrm{20} \\ $$$$=\mathrm{5}\:\mathrm{ln}\:\mathrm{2} \\ $$

Commented by mokys last updated on 27/Jan/23