Question

If \( f'(x) = x^2 + 5 \) and \( f(0) = -1 \) then \( f(x) = ..............\)

Hint:

This type of problem is called an initial value problem. The process is always the same: integrate to find the general solution with a constant \(C\), then use the given point (the initial value) to solve for \(C\) and find the particular solution.

Answer 1

Step 1: Find the general form of \( f(x) \) by integrating \( f'(x) \) with respect to \( x \). \[ f(x) = \int (x^2 + 5) dx = \frac{x^3}{3} + 5x + C \] where \( C \) is the constant of integration.
Step 2: Use the given initial condition, \( f(0) = -1 \), to find the value of \( C \). \[ f(0) = \frac{0^3}{3} + 5(0) + C = -1 \] \[ C = -1 \] Step 3: Substitute the value of \( C \) back into the expression for \( f(x) \). \[ f(x) = \frac{x^3}{3} + 5x - 1 \]