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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
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
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