how can i integrate log sinx from 0 to pi/2 . plz give full stepwise solution

Dude do integration by parts.....................u=1 and v=f(x) then properly substitute sinx if reqd..............

Dear Bharat

Let I =limit  o to pi/2  ∫ log sinx dx ....................1

use property

I= limit 0 to pi     ∫log sin(pi/2 -x) dx

or

I =

I= limit 0 to pi/2     ∫log cos(x) dx..................2

1+2

2I =limit 0 to pi /2    ∫log sinxcos(x) dx

2I =limit 0 to pi/2     ∫log (sin2x)/2  dx

2I =limit 0 to pi/2     ∫log sin2x dx   -  limit 0 to pi/2 ∫log2 dx

2I =limit 0 to pi /2    ∫log sin2x dx   -  pi/2*log2........................3 =

let I₁=imit 0 to pi/2     ∫log sin2x dx

    2x=t 

  2dx=dt

 I₁=imit 0 to ∏     1/2∫log sint dt

use property

I₁=imit 0 to ∏/2     2*1/2∫log sint dt

I₁=imit 0 to ∏/2     log sint dt    =I

put this in equation 3

2I=I -∏/2  log2

I=-∏/2 log2