\( x^{3}-\frac{1}{x^{2}} \). Find the equation of \( C \) in the form \( y=f(x) \). The gradient of a particular curve is given by \( \frac{d y}{d x}=\frac{\sqrt{x}+3}{x^{2}} \). Given thrt the point ( 9,0 ). ind the equation of the curve.

To find the equation of the curve \( C \) given the gradient \( \frac{dy}{dx} = \frac{\sqrt{x} + 3}{x^2} \), we need to integrate the gradient function with respect to \( x \). ### Step 1: Integrate the gradient We have: \[ \frac{dy}{dx} = \frac{\sqrt{x} + 3}{x^2} \] We can rewrite this as: \[ \frac{dy}{dx} = \frac{\sqrt{x}}{x^2} + \frac{3}{x^2} = \frac{x^{1/2}}{x^2} + \frac{3}{x^2} = \frac{1}{x^{3/2}} + \frac{3}{x^2} \] Now, we integrate each term separately: \[ y = \int \left( \frac{1}{x^{3/2}} + \frac{3}{x^2} \right) dx \] ### Step 2: Integrate each term 1. For \( \int \frac{1}{x^{3/2}} \, dx \): \[ \int x^{-3/2} \, dx = \frac{x^{-1/2}}{-1/2} = -2x^{-1/2} = -\frac{2}{\sqrt{x}} \] 2. For \( \int \frac{3}{x^2} \, dx \): \[ \int 3x^{-2} \, dx = 3 \cdot \frac{x^{-1}}{-1} = -\frac{3}{x} \] ### Step 3: Combine the results Combining the results of the integrals, we have: \[ y = -\frac{2}{\sqrt{x}} - \frac{3}{x} + C \] where \( C \) is the constant of integration. ### Step 4: Use the point (9, 0) to find \( C \) We know that the curve passes through the point \( (9, 0) \). We can substitute \( x = 9 \) and \( y = 0 \) into the equation to find \( C \): \[ 0 = -\frac{2}{\sqrt{9}} - \frac{3}{9} + C \] Calculating the terms: \[ 0 = -\frac{2}{3} - \frac{1}{3} + C \] \[ 0 = -1 + C \] Thus, we find: \[ C = 1 \] ### Step 5: Write the final equation Substituting \( C \) back into the equation for \( y \): \[ y = -\frac{2}{\sqrt{x}} - \frac{3}{x} + 1 \] Thus, the equation of the curve \( C \) in the form \( y = f(x) \) is: \[ \boxed{y = -\frac{2}{\sqrt{x}} - \frac{3}{x} + 1} \]