Find f(x) if f'(x)=11e^x/(11+e^x).

The function we have to find out is called primitive and
it is noted as F(x). To determine F(x), we'll solve the indefinite integral using
substitution technique:

11+e^x =
t

We'll differentiate both
sides:

e^xdx = dt

Int
11e^xdx/(11+e^x)  = 11*Int dt/t

11*Int dt/t = 11*ln |t| +
C

We'll replace t by 11+e^x and we'll apply power property
of logarithms:

F(x) = Int 11e^xdx/(11+e^x)  =
ln [(11+e^x)^11] + C