Question:

Consider the function f(x) defined by: f(x) = ∫(2x^3 - x^2 + 4)dx

a) Find the indefinite integral of f(x).

b) Evaluate the definite integral ∫[1, 3] (2x^3 - x^2 + 4)dx.

Answer:

a) To find the indefinite integral, we need to apply the power rule of integration.

∫(2x^3 - x^2 + 4)dx = (2/4)x^4 - (1/3)x^3 + 4x + C

Therefore, the indefinite integral of f(x) is given by: F(x) = (1/2)x^4 - (1/3)x^3 + 4x + C

b) To evaluate the definite integral, we can directly substitute the upper and lower limits into the indefinite integral and subtract their results.

∫[1, 3] (2x^3 - x^2 + 4)dx = F(3) - F(1)

Substituting the limits into the indefinite integral expression:

= [(1/2)3^4 - (1/3)3^3 + 4(3)] - [(1/2)1^4 - (1/3)1^3 + 4(1)]

Simplifying,

= [(1/2)81 - (1/3)27 + 12] - [(1/2)1 - (1/3) + 4]

= [40.5 - 9 + 12] - [0.5 - 0.33 + 4]

= [43.5] - [3.83]

= 39.67

Hence, the value of the definite integral ∫[1, 3] (2x^3 - x^2 + 4)dx is approximately 39.67.