prove-that-0-pi-2-ln-1-1-2-sin2x-dx-

Question Number 124177 by mathdave last updated on 01/Dec/20

p r o v e t h a t

\int_{0}^{\frac{π}{2}} \ln (1 - \frac{1}{2} \sin 2 x) d x

Commented by MJS_new last updated on 01/Dec/20

prove what ?

Commented by Dwaipayan Shikari last updated on 01/Dec/20

\int_{0}^{1} l o g (1 - t \sqrt{1 - t^{2}})

= \int_{0}^{1} \frac{l o g (1 - t \sqrt{1 - t^{2}})}{\sqrt{1 - t^{2}}} d t

= - \underset{n = 1}{\sum^{\infty}} \frac{1}{n} \int_{0}^{1} t^{n} {(1 - t^{2})}^{\frac{n - 1}{2}} d t t^{2} = u \Rightarrow 2 t = \frac{d u}{d t}

= - \frac{1}{2} \underset{n = 1}{\sum^{\infty}} \frac{1}{n} \int_{0}^{1} u^{\frac{n}{2} - \frac{1}{2}} {(1 - u)}^{\frac{n - 1}{2}} d u

= - \frac{1}{2} \underset{n = 1}{\sum^{\infty}} \frac{Γ^{2} (\frac{n + 1}{2})}{n Γ (n + 1)}

= - \frac{1}{2} \underset{n = 1}{\sum^{\infty}} \frac{Γ^{2} (\frac{n + 1}{2})}{n^{2} Γ (n)} = - 0.6403