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 =