Calculus: Early Transcendentals, Chapter 5, 5.4, Section 5.4, Problem 22

You need to evaluate the integral, hence, you need to use the fundamental theorem of calculus, such that:
int_a^b f(x)dx = F(b) - F(a)
int_1^2(4x^3 - 3x^2 + 2x)dx = int_1^2(4x^3)dx - int_1^2 3x^2 dx + int_1^2 2x dx
Evaluating each definite integral, using the formula int x^n dx = (x^(n+1))/(n+1) + c , yields:
int_1^2(4x^3)dx = 4*(x^4)/4|_1^2 = 2^4 - 1^4 = 15
int_1^2 3x^2 dx =3x^3/3|_1^2 = 2^3 - 1^3 = 7
int_1^2 2x dx = 2x^2/2|_1^2 = 2^2 - 1^2 = 3
Gathering the results, yields:
int_1^2(4x^3 - 3x^2 + 2x)dx = 15 - 7 + 3 = 11
Hence, evaluating the definite integral, using the fundamental theorem of calculus yields int_1^2(4x^3 - 3x^2 + 2x)dx = 11.