Riley hobson bence · Problem 5.33

Partial differentiation

Mathematics — Methods · differentiation-under-integral-sign

Problem

I(\alpha) = \int_0^1 \frac{x^\alpha - 1}{\ln x}\, dx, \qquad \alpha > -1,

what is the value of

I(0)

? Show that

\frac{d}{d\alpha} x^\alpha = x^\alpha \ln x,

and deduce that

\frac{d}{d\alpha} I(\alpha) = \frac{1}{\alpha + 1}.

Hence prove that

I(\alpha) = \ln(1 + \alpha)