Calculus: Early Transcendentals, Chapter 4, 4.9, Section 4.9, Problem 37

You need to evaluate f and the problem provides f'(x), hence, you need to use the following relation, such that:
int f'(x)dx = f(x)+ c
You need to evaluate the indefinite integral of the power function, hence, you need to use the following formula:
int x^(-n) dx = (x^(-n+1))/(-n+1) + c
int x^(-1/3) dx = (x^(-1/3+1))/(-1/3+1) + c
int x^(-1/3) dx =(3/2)*x^(2/3) + c
Hence,f(x) = (3/2)*x^(2/3) + c . You may find c using the following information, such that:
f(1) =1 => f(1) = (3/2)*1^(2/3) + c => 3/2 + c = 1 => c = 1 -3/2 => c = -1/2
Hence, evaluating f(x) under the given condition, yields f(x) = (3/2)*root(3)(x^2) - 1/2.