Question

Please show me how to do this.

Answer 1

`f(x) = (1)/(2)(x - \ln (x - 1) - 2)`

Explanation:

`f'(x) = (1)/(2) - (1)/(2x - 2)`

`f(x) = \int ((1)/(2) - (1)/(2x - 2)) dx`

`f(x) = \int [(1)/(2) - (1)/(2(x - 2))] dx`

`f(x) = \int (1)/(2)dx - (1)/(2) \int (1)/(x - 1) dx`

`f(x) = (1)/(2)x - (1)/(2) \ln (x - 1) + C`

`f(x) = (x - \ln (x - 1))/(2) + C`

`f(2) = (2 - \ln (2 - 1))/(2) + C = 0`

`f(2) = (2 - \ln (1))/(2) + C = 0`

`f(2) = (2)/(2) + C = 0`

`f(2) = 1 + C = 0`

`C = -1`

`f(x) = (x - \ln (x - 1))/(2) - 1`

`f(x) = (1)/(2)[x - \ln (x - 1) - 2]`