Question

Show that

∫ₐᵇ (ln x)/(x) dx = (1/2) ln ab ln(a/b)

Answer 1

$\intop ^b_a \frac {ln(x)} x dx$

Find

$\int \frac {ln(x)} x dx = \int ln(x) d(ln(x))=[ln(x)=t]=\int t dt = \frac {t^2} 2 + C = \frac {ln^2(x)} 2 + C$

than

$\intop ^b_a \frac {ln(x)} x dx = \frac {ln^2(x)} 2|^b _a = \frac {ln^2(b)} 2 - \frac {ln^2(a)} 2 = \frac 1 2 (ln^2 (b) - ln^2(a)) = \frac 1 2 (ln(b) - ln(a))(ln(b) +ln(a)) = \frac 1 2 (ln (\frac b a)ln(ab))$