prove-the-convergence-of-0-1-ln-1-x-x-dx-

Question Number 74793 by mathmax by abdo last updated on 30/Nov/19

p r o v e t h e c o n v e r g e n c e o f \int_{0}^{1} \frac{l n (1 + \sqrt{x})}{\sqrt{x}} d x

Commented by mathmax by abdo last updated on 06/Dec/19

I = \int_{0}^{1} \frac{l n (1 + \sqrt{x})}{\sqrt{x}} d x c h a n g e m e n t \sqrt{x} = t g i v e x = t^{2} \Rightarrow

I = \int_{0}^{1} \frac{l n (1 + t)}{t} (2 t) d t = 2 \int_{0}^{1} l n (1 + t) d t =_{1 + t = u} 2 \int_{1}^{2} l n (u) d u

= 2 {[u l n u - u]}_{1}^{2} = 2 {2 l n (2) - 2 + 1} = 4 l n (2) - 2

Answered by mind is power last updated on 01/Dec/19

u = \sqrt{x} \Rightarrow du = \frac{1}{2 \sqrt{x}} dx

\int_{0}^{1} 2 \ln (1 + u) du = {[2 (u + 1) \ln (u + 1) - 2 u]}_{0}^{1} = 4 \ln (2) - 2