Question

For x > 0, let f(x) = ∫(ln t/(1 + t)) dt for t ∈ [1,x]. 

Find the function f(x) + f (1/x) and show that f(e) + f (1/e) = 1/2. 

Here, ln t = log_et.

Answer 1

We have

Now,

Let 

t = 1/y 

dt = – (1/y²) dy

The given integral reduces to

Adding Eqs (i) and (ii), we get 

 

Thus, f(e) + f(1/e) 

= (ln e/2)² 

= 1/2