Calculus: Early Transcendentals, Chapter 4, 4.9, Section 4.9, Problem 32

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
int (5x^4 - 3x^2 + 4)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 5x^4dx = 5(x^(4+1))/(4+1) + c => int 5x^4dx = x^5 + c
int 3x^2 dx =x^3 + c
int 4dx = 4x + c
Hence,f(x) = x^5 - x^3 + 4x + c . You may find c using the following information, such that:
f(-1) =2 => f(-1) = (-1)^5 - (-1)^3 + 4*(-1) + c => -1 + 1 - 4 + c = 2 => c = 6
Hence, evaluating f(x) under the given condition, yields f(x) = x^5 - x^3 + 4x + 6 .