Q.358. Definite Integration.

Question 358.
∫ (x = ln2 to ∞) dx / (e^x - 1).

Answer 358.
1/(e^x - 1) dx
= ∫ e^-x / (1 - e^-x) dx
Let 1 - e^-x = t => e^-x dx = dt
Also, x = ln 2 => t = 1 - e^(-ln2) = 1/2
and x = ∞ => t = 1
=> Integral
= ∫ dt/t
= ln l t l + c
Plugging limits t = 1/2 to 1
= ln1 - ln(1/2)
= ln2.

Link to YA!