The curve with equation y = f(x) passes through the point (1, 6) - Edexcel - A-Level Maths Pure - Question 10 - 2006 - Paper 1

To find f(x), we start with the expression for f'(x):

$f'(x) = 3 + \frac{5x^2 + 2}{x^3}$

We can rewrite this by finding a common denominator:

$f'(x) = 3 + \frac{5x^2 + 2}{x^3} = 3 + \frac{5}{x} + \frac{2}{x^3}$

Next, we integrate f'(x):

$f(x) = \int (3 + \frac{5}{x} + \frac{2}{x^3}) \, dx$

Carrying out the integration term by term, we have:

• The integral of 3 is 3x.
• The integral of \frac{5}{x} is 5 \ln |x|.
• The integral of \frac{2}{x^3} is -\frac{2}{2x^2} = -\frac{1}{x^2}.

Thus,

$f(x) = 3x + 5 \ln |x| - \frac{1}{x^2} + C$

where C is the constant of integration.