Evaluate the following integral:

Let us assume….....equation 1


By property, we know that


.....………equation 2


Adding equation 1 and equation 2



We know



……….equation 3


We know


if f(2a – x) = f(x)


= 0 if f(2a – x) = – f(x)


Thus equation 3 becomes


………equation 4 since logsin(π – x) = logsinx


By property, we know that



………equation 5


Adding equation 4 and equation 5


+


We know




We know logm + logn = logmn thus




since log(m/n) = logm – logn


.....equation 6


Let


Let 2x = y


2dx = dy


dx = dy/2


For x = 0


y = 0


for


y = π


thus substituting value in I1



From equation 3 we get




Thus substituting the value of I1 in equation 6







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