Evaluate the following integral:
Let us assume
since logmn = logm + logn and log(m)n = nlogm
.......equation (a)
Let
We know that
If f(2a – x) = f(x)
than
thus
………equation 1
since logsin(π – x) = logsinx
By property, we know that
………equation 2
Adding equation 1 and equation 2
+
We know
We know logm + logn = logmn thus
since log(m/n) = logm – logn
.....equation 3
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 I2 in equation 3
Substituting in equation (a) i.e
I =