A curve has equation y = f(x) - Edexcel - A-Level Maths Pure - Question 11 - 2013 - Paper 2

To find f(x), we need to integrate f'(x).

1. Start with: $f'(x) = \frac{x + 9}{\sqrt{x}}$

2. Simplify the expression by splitting it into two fractions: $f'(x) = \frac{x}{\sqrt{x}} + \frac{9}{\sqrt{x}} = \sqrt{x} + 9x^{-\frac{1}{2}}$

3. Now integrate each term: $f(x) = \int (\sqrt{x} + 9x^{-\frac{1}{2}}) dx$

   • For $\sqrt{x}$, the integral yields: $\frac{2}{3}x^{\frac{3}{2}}$
   • For $9x^{-\frac{1}{2}}$, the integral yields: $9 \cdot 2x^{\frac{1}{2}} = 18\sqrt{x}$

4. Combine the results: $f(x) = \frac{2}{3}x^{\frac{3}{2}} + 18\sqrt{x} + C$

5. To find the constant C, use the point P(9, 0):

   • Substitute x = 9 into f(x): $0 = \frac{2}{3}\cdot 9^{\frac{3}{2}} + 18\cdot 9^{\frac{1}{2}} + C$ $0 = \frac{2}{3} \cdot 27 + 18 \cdot 3 + C$ $0 = 18 + 54 + C$ $C = -72$

6. Thus, the final function is: $f(x) = \frac{2}{3}x^{\frac{3}{2}} + 18\sqrt{x} - 72$