Skip to content
Let $I = \int_0^\pi x \log \sin xdx$ ……..(i)
$I = \int_0^\pi {(\pi – x)\log \sin (\pi – x)} dx$
$ = \int_0^\pi {(\pi – x)} \log \sin xdx$ …….(ii)
$2I = \pi \int_0^\pi {\log } \sin xdx$ …….(iii)
$2I = 2\pi \int_0^{\pi /2} {\log } \sin xdx$
$I = \pi \int_0^{\pi /2} {\log } \sin xdx$ ………(iv)
Now, $I = \pi \int_0^{\pi /2} {\log } \sin (\pi /2 – x)dx$ …….(v)
On adding Eqs. (iv) and (v), we get $2I = \pi \int_0^{\pi /2} {(\log \sin x + \log \cos x)} dx$
$2I = \pi \int_0^{\pi /2} {\log } \sin x\cos xdx$
$ = \pi \int_0^{\pi /22} {\log } \frac{{2\sin x\cos x}}{2}dx$
$2I = \pi \int_0^{\pi /2} {(\log \sin 2x – \log 2)} dx$
$2I = \pi \int_0^{\pi /2} {\log } \sin 2xdx – \pi \int_0^{\pi /2} {\log } 2dx$
Let’s put $2x = t \Rightarrow dx = \frac{1}{2}dt$
As $x \to 0,$ then $t \to 0$
and $x \to \frac{\pi }{2},$ then $t \to \pi $
therefore,$2I = \frac{\pi }{2}\int_0^\pi {\log } \sin tdt – \frac{{{\pi ^2}}}{2}\log 2$
$ \Rightarrow $$2I = \frac{\pi }{2}\int_0^\pi {\log } \sin xdx – \frac{{{\pi ^2}}}{2}\log 2$
$ \Rightarrow $$2I = I – \frac{{{\pi ^2}}}{2}\log 2$ [from eq.(iii)]
therefore,$I = – \frac{{{\pi ^2}}}{2}\log 2 = \frac{{{\pi ^2}}}{2}\log \left( {\frac{1}{2}} \right)$
Purchase Best Mathematics – E-Books from : https://mathstudy.in/