Question

Evaluate: `int_0^pi e^cosx/(e^cos x + e^-cos x) dx`

Answer 1

Let I = `int_0^pi e^cos x/(e^cos x + e^-cos x) dx`    ...(i)

∴ I = `int_0^pi e^cos(pi-x)/(e^cos(pi-x) + e^-cos(pi - x)) dx      ...["Using" int_0^a f(x) dx = int_0^a f(a - x)dx]` 

or, I = `int_0^pi e^-cos x/(e^-cos x + e^ cos x)` dx    ...(ii)

On adding eqs. (i) & (ii), we get 

2I = `int_0^pi (e^cos x + e^-cos x)/(e^cos x + e^-cos x) dx`

= `int_0^pi dx`

= `[x]_0^pi = pi`

∴ I = `(pi)/2`