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

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

Grade:Upto college level

3 Answers

manas pawar
23 Points
11 years ago

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

Badiuddin askIITians.ismu Expert
147 Points
11 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

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


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Sudipto Koley
32 Points
5 years ago
∫(a+b sin x) / (b+a sin x) dx

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