Question

Question:

Evaluate the following integral using Gamma function.

Integrate x from 0-->1 dx/((-ln(x))^(1/2))


My approach:

I let -ln(x) = u ==> ln(x) = -u ==> x = e^(-u)

==> dx = -(e^(-u))

Limits of u: 0-infinty, which gives,



Integrate u from 0-->1 dx/((-ln(x))^(1/2)) = Integrate u from 0-->infinity { - du/((u^(1/2))*(e^(-u))) }

= -Gamma function[ (-1/2) + 1 ] = -(-1/2) * Gamma function [ (-1/2) ]

= (1/2) * (-2 * sqrt(pi) )

= -sqrt(pi)

Is this approach correct?

Thank you in advance!

Answer 1

looks good, but I get √π
did you lose track of the sign somewhere?