Question

If f'(x) = x² + 5 and f(0) = −1 then f(x) = ______.

Answer 1

If f'(x) = x² + 5 and f(0) = −1 then f(x) = `bbunderline(x^3/3 + 5x - 1)`.

Explanation:

The function f(x) is obtained by integrating f′(x): f'(x) = x² + 5

Step 1: Integrate f'(x)

`f(x) = int(x^2 + 5)dx = x^3/3 + 5x + C`

Step 2: Apply initial condition f(0) = −1

f(0) = `0^3/3` + 5(0) + C = -1

C = −1

Final Function:

`f(x) = x^3/3 + 5x - 1`