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

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

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

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

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giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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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