calculate int_0^pi∫_0^x log(sin(x−y))dydx I was asked to find the integral int int_A log(sin(x−y))dxdy where A is the triangle y=0,x=pi,y=x in the first quadrant.

lollaupligey9

lollaupligey9

Answered question

2022-08-11

calculate 0 π 0 x log ( sin ( x y ) ) d y d x
I was asked to find the integral A log ( sin ( x y ) ) d x d y where A is the triangle y = 0 , x = π , y = x in the first quadrant.
I was given a hint: evaluate 0 π log ( sin ( t ) ) d t using symmetry.
What I did:
I inferred from the hint that the variable change t = x y is the way to go, so t = x y, and since we integrate by y first, then x is a "constant" and d t = d y, and since y transitions from 0 to x, then t transitions from x to 0, so we can rewrite the integral:
0 π 0 x log ( sin ( x y ) ) d y d x = 0 π x 0 log ( sin ( t ) ) d t d x = 0 π 0 x log ( sin ( t ) ) d t d x
And here I am stuck. Firstly, I don't know how 0 π log ( sin ( t ) ) d t is related to the question, since in the question the limits are 0 and x. not 0 and π. They are not the same thing (even though x transitions from 0 to π).
But even if I did, how would I evaluate 0 π log ( sin ( t ) ) d t??

Answer & Explanation

Avah Leonard

Avah Leonard

Beginner2022-08-12Added 21 answers

At the point
0 π 0 x log ( sin t ) d t d x ,
changing the order of integration is a tempting thing to do:
0 π 0 x log ( sin t ) d t d x = 0 π t π log ( sin t ) d x d t (a) = 0 π ( π t ) log ( sin t ) d t = 0 π u log ( sin ( π u ) ) d u (b) = 0 π u log ( sin u ) d u .
Now add ( a ) and ( b ), and use the symmetry.
muroscamsey

muroscamsey

Beginner2022-08-13Added 3 answers

0 π log ( sin x ) d x = 2 0 π / 2 log ( sin x ) d x = 0 π / 2 log ( sin x ) d x + 0 π / 2 log ( cos x ) d x = 0 π / 2 log ( sin x cos x ) d x = 0 π / 2 [ log ( sin 2 x ) log ( 2 ) ] d x = 1 2 0 π log ( sin x ) d x π log ( 2 ) / 2
Thus
0 π log ( sin x ) d x = π log ( 2 )

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