Question

Let E(x) = integral of (e^(x))/(e^(x)+e^(-x)) and F(x) = integral of (e^(-x))/(e^(x)+e^(-x)). 

(a) Calculate E(x) + F(x)

 
(b) Calculate E(x) - F(x)

 
(c) Use your results from parts (a) and (b) to calculate E(x) and F(x).

Answer 1

(a)

The functions are and

Then E(x) + F(x)

= x

Therefore E(x) + F(x) = x

Answer 2

(b)

The Functions are and

Then E(x) - F(x)

Let us assume

 = ln u                            (Using Logarithmic Rule )

Therefore 

Answer 3

(c)

The Functions are and

E(x) + F(x) = x and

Add E(x) + F(x) with E(x) - F(x).

E(x) + F(x) + E(x) - F(x) =

Similarly Subtract  from E(x) + F(x) = x

E(x) + F(x) - E(x) + F(x) =

Therefore and .