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