What is `int (e^(x) + 1)^(-1) dx` equal to?

Options

A. `ln (e^(x) + 1) + c`
B. `ln (e^(-x) + 1) + c`
C. `-l (e^(-x) + 1) + c`
D. `-(e^(x) + 1) + c`

Answer

C

Explanation

Let I `= int (e^(x) + 1)^(-1) dx = int (1)/(e^(x) + 1) dx = int (e^(-x))/(1 + e^(-x)) dx`
Let `1 + ^(-x) =t rArr -e^(-x) dx = dt`
`:. I = - int (1)/(t) dt = - log t + c = - log (1 + e^(-x)) + c`