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

bharat bhushan sharma , 15 Years ago

Grade Upto college level

3 Answers

manas pawar

Last Activity: 15 Years ago

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

Badiuddin askIITians.ismu Expert

Last Activity: 15 Years ago

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

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

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