To integrate the expression (4x^3 + √x) / 2x^2, you can split it into two separate integrals and then solve each integral separately. Let's break it down step by step:

  1. Separate the terms: (4x^3 + √x) / 2x^2 = (4x^3 / 2x^2) + (√x / 2x^2)

  2. Simplify each term: (4x^3 / 2x^2) = 2x (√x / 2x^2) = (1 / 2x^(3/2))

  3. Integrate each simplified term: ∫(2x) dx = x^2 + C1 ∫(1 / 2x^(3/2)) dx = ∫(1/2) * x^(-3/2) dx = (1/2) * (2 / (-1/2)) * x^(-1/2) + C2 = -x^(-1/2) + C2 Here, C1 and C2 are constants of integration.

  4. Combine the results: The final result of integrating the original expression is: ∫[(4x^3 + √x) / 2x^2] dx = x^2 - x^(-1/2) + C, where C is the constant of integration.

Therefore, the integral of (4x^3 + √x) / 2x^2 is x^2 - x^(-1/2) + C, where C is the constant of integration.