Question

What is `f(x) = int (3-x)/(x-4)` if `f(5)=1`?

Answer 1

`f(x) = 6 - x - ln|x- 4|`

Explanation:

`f(x) = int(-(x - 3))/(x- 4)dx`

`f(x) = -int(x - 3)/(x - 4)dx`

Decompose into partial fractions.

`A/1 + B/(x -4) = (x - 3)/(1(x - 4))`

`A(x- 4) + B(1) = x - 3`

`Ax - 4A + B = x - 3`

Write a system of equations:

`{(A = 1), (B - 4A = -3):}`

Solve:

`A = 1` and `B = 1`.

`f(x) =- int 1 + 1/(x - 4)dx`

This can be integrated using `int(1/x) = ln|x| + C` and `int(x^n)dx = x^(n +1)/(n + 1) + C`.

`f(x) = -x - ln|x- 4| + C`

Solve for `C` now.

`1 = -5 - ln|5 - 4| + C`

`1 = -5 + 0 + C`

`C = 6`

The function is therefore `f(x) = 6 - x - ln|x- 4|`.

Hopefully this helps!