allery$\displaystyle\int_0^1 \dfrac{x - 1}{\ln x}\, dx = ?$ (Frullani-type)
A$0$
B$\ln 2$
C$1$
D$e$
Answer & Solution
Correct answer: B. $\ln 2$
Use Feynman's trick.
Define $I(s) = \int_0^1 (x^s - 1)/\ln x\, dx$.
Then $I'(s) = \int_0^1 x^s\, dx = 1/(s+1)$. $I(0) = 0$.
**Integrate:** $I(s) = \ln(s+1)$.
For our problem, $s = 1$: $I(1) = \ln 2$.
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