What is antiderivative of function f(x) = ln(x-1)/(x-1)?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find the anti-derivative of the function `f(x)` , hence, you need to evaluate the indefinite integral of the given function, such that:

`int f(x)dx = int (ln(x-1))/(x-1)dx`

You should use the substitution process, such that:

`ln(x-1) = t => 1/(x-1)*dx = dt`

Replacing the variable yields:

`int (ln(x-1))/(x-1)dx = int t*dt`

`int t*dt = t^2/2 + c`

Replacing back `ln(x-1)` for `t` yields:

`int (ln(x-1))/(x-1)dx = (ln(x-1))^2/2 + c`

Hence, evaluating the anti-derivative of the given function yields `int f(x)dx = (ln(x-1))^2/2 + c.`

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