Calculus: Early Transcendentals, Chapter 4, 4.9, Section 4.9, Problem 36

You need to evaluate the antiderivative of the function f'(x), such that:
int f'(x)dx = f(x) + c
int (x^2-1)/x dx = int x^2/x dx - int 1/x dx
int (x^2-1)/x dx = int x dx - int 1/x dx
int (x^2-1)/x dx = x^2/2 - ln|x| + c
Hence, f(x) = x^2/2 - ln|x| + c.
The function is indeterminate, because of the constant c, but the problem provides the information that f(1) = 1/2 and f(-1) = 0, hence, you may evaluate the constant c, such that:
f(1) = 1/2 - ln 1 + c => 1/2 = 1/2 - 0 + c => c = 0
f(-1) = 1/2 - ln 1 + c => 0 = 1/2 - 0 + c => c = 1/2
Hence, evaluating the function yields f(x) = x^2/2 - ln|x| for x >= 1 and f(x) = x^2/2 - ln|x| + 1/2 , for x <= -1.