Integrate log (sin x) from 0 to pi /2?

We need to Integrate log(sinx) from 0 to pi /2

Solution

Let I =limit o to pi/2 ∫ log sinx dx —(i)

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———–(ii)

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———–(iii)

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

Answer

-∏/2 log2

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