Question

$$\int_0^\pi\,\ln(\sin{x})\,\sqrt[n]{\csc{x}}\:dx$$

İntegralini çözün.

Answer 1

İntegralimiz :

$$\int_0^\pi\,\ln(\sin{x})\,\sqrt[n]{\csc{x}}\:dx$$

Buradaki integralin $n$ 'e göre türevini alalım.

$$\frac{\partial}{\partial{n}}\int_0^\pi\:\sqrt[n]{\csc(x)}\:dx=\frac{\partial}{\partial{n}}\frac{\Gamma^2\Big(\frac{1}{2}-\frac{1}{2n}\Big)}{\sqrt[n]{2}\:\Gamma\Big(1-\frac{1}{n}\Big)}$$

$$\color{#A00000}{\boxed{\int_0^\pi\,\ln(\sin{x})\,\sqrt[n]{\csc{x}}\:dx=\frac{\Gamma^2\Big(\frac{1}{2}-\frac{1}{2n}\Big)}{\sqrt[n]{2}\:\Gamma\Big(1-\frac{1}{n}\Big)}\Bigg[\ln(2)+\psi\bigg(\frac{1}{2}-\frac{1}{2n}\bigg)-\psi\bigg(1-\frac{1}{n}\bigg)\Bigg]}}$$