Given-that-0-12-f-x-dx-20-find-the-value-of-1-8-f-4-log-2-x-x-dx-

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

Given that \int_{0}^{12} f (x) d x = 20,

find the value of \int_{1}^{8} \frac{f (4 \log_{2} x)}{x} d x .

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

l e t u = {4 \log}_{2} x

x = 2^{\frac{u}{4}}

d x = \frac{\ln 2}{4} \times 2^{\frac{u}{4}} d u

\int_{1}^{8} \frac{f (4 \log_{2} x)}{x} d x = \int_{0}^{12} \frac{f (u)}{2^{\frac{u}{4}}} \times \frac{\ln 2}{4} \times 2^{\frac{u}{4}} d u

= \frac{\ln 2}{4} \int_{0}^{12} f (u) d u = 5 \ln 2

Commented by ZiYangLee last updated on 13/Sep/21

w o w t h a n k s m r W