# Find the antiderivative of the function given by f(x)=(ln(2x-5))/(2x-5)?

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to find the anti derivative of f(x) = (ln(2x-5)) / (2x-5).

This can be done using substitution

Int [ (ln(2x-5)) / (2x-5) dx]

let ln(2x - 5)  = t

=> dt/dx = 2/(2x - 5)

=> dt / 2 = dx/(2x - 5)

Int [ (ln(2x-5)) / (2x-5) dx]

=> Int [ t/2  dt]

=> t^2/4 + C

substitute t = ln (2x - 5)

=> [ln (2x - 5)]^2 / 4 + C

The anti derivative of f(x) = (ln(2x-5)) / (2x-5) is [ln (2x - 5)]^2 / 4 + C

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

To determine the antiderivative of the given function, we'll have to evaluate the indefinite integral of the function.

We'll replace 2x-5 by t.

ln(2x-5) = t

We'll differentiate both sides and we'll get:

2dx/(2x-5)= dt

dx/(2x-5)=dt/2

We'll re-write the integral in the changed variable:

Int [ln (2x-5)] dx/ (2x-5)  = Int t*dt/2

Int t*dt/2 = t^2/4 + C

The primitive function is: F(x) =  [ln(2x-5)]^2/4 + C