Question

What is `f(x) = int (x^2+x)/((x+1)(x-1) ) dx` if `f(3)=2`?

Answer 1

`f(x) = x+ ln|x-1|-1-ln(2)`

Explanation:

Given: `f(x) = int (x^2+x)/((x+1)(x-1) ) dx, f(3) = 2`

Factor the numerator of the integrand:

`f(x) = int (x(x+1))/((x+1)(x-1) ) dx, f(3) = 2`

Cancel `(x+1)/(x+1) to 1`:

`f(x) = int x/(x-1) dx, f(3) = 2`

Add 0 to the numerator in the form of `-1+1`:

`f(x) = int (x-1+1)/(x-1) dx, f(3) = 2`

Separate into two fractions:

`f(x) = int (x-1)/(x-1)+1/(x-1) dx, f(3) = 2`

The first fraction becomes 1:

`f(x) = int 1+1/(x-1) dx, f(3) = 2`

The integrals of the two terms are well know:

`f(x) = x+ln|x-1|+C, f(3) = 2`

Evaluate at `x = 3`

`2 = 3+ln|3-1|+C, f(3) = 2`

Solve for C:

`C = -1-ln(2)`

This is function that satisfies the boundary condition, `f(3) = 2`:

`f(x) = x+ ln|x-1|-1-ln(2)`