Question

$$\int_0^1\:\frac{\ln(x)\:\Big[1+x^{-\frac{1}{3}}\Big]}{(1-x)\sqrt[3]{x}}\:dx$$

İntegralini çözün.

Answer 1

Buradaki eşitlikte $s$ yerine $\frac{1}{3}$ koyarsak integrali buluruz.

$$\lim\limits_{s\to\frac{1}{3}}\:\int_0^1\:\frac{\ln(x)\big(x^{s-1}+x^{-s}\big)}{(1-x)}\:dx=\int_0^1\:\frac{\ln(x)\:\Big[1+x^{-\frac{1}{3}}\Big]}{(1-x)\sqrt[3]{x}}\:dx$$

$$\large\color{#A00000}{\boxed{\int_0^1\:\frac{\ln(x)\:\Big[1+x^{-\frac{1}{3}}\Big]}{(1-x)\sqrt[3]{x}}\:dx=-\frac{4}{3}\pi^2\approx-13.159472}}$$