To integrate x² - √x/x with respect to x, we can start by factoring the expression as follows:
x² - √x/x = x - 1/√x
Then we can integrate each term separately:
∫(x - 1/√x) dx = ∫x dx - ∫(1/√x) dx
The first integral is straightforward:
∫x dx = 1/2 x² + C1
For the second integral, we can use the substitution u = √x, du/dx = 1/(2√x), dx = 2√x du:
∫(1/√x) dx = ∫2 du = 2u + C2 = 2√x + C2
Substituting back u = √x, we get:
∫(1/√x) dx = 2√x + C2
Putting everything together, we have:
∫(x² - √x/x) dx = ∫x dx - ∫(1/√x) dx = (1/2 x² + C1) - (2√x + C2)
= 1/2 x² - 2√x + C
where C = C1 - C2 is the constant of integration.
Therefore, the correct option is x²/2 - 2√x + K, where K = C is the constant of integration.