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

Expert Answers
justaguide eNotes educator| Certified Educator

We have to find the integral of f'(x)=11e^x/(11+e^x)


let 11 + e^x = y

e^x dx = dy

Int [ 11e^x/(11+e^x) dx]

=> Int [ 11dy/y]

=> 11*ln |y| + C

substitute y = 11 + e^x

=> 11*ln |(11 + e^x)| + C

For f'(x) = 11e^x/(11+e^x), f(x) = 11*ln |(11 + e^x)| + C

giorgiana1976 | Student

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

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