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