Question

$$\int_0^1\:\frac{\ln{x}}{(1-x)\sqrt{x}}\:dx$$

İntegralini çözün.

Answer 1

Buradaki eşitlikte , $s$ yerine $\frac{1}{2}$ koyalım.

$$\lim\limits_{s\to\frac{1}{2}}\:\int_0^1\:\frac{\ln(x)\big(x^{s-1}+x^{-s}\big)}{(1-x)}\:dx=2\int_0^1\:\frac{\ln{x}}{(1-x)\sqrt{x}}\:dx$$

$$\large\color{#A00000}{\boxed{\int_0^1\:\frac{\ln{x}}{(1-x)\sqrt{x}}\:dx=-\frac{\pi^2}{2}\approx-4.934802}}$$