Find f(x) if f'(x) = x^2 -5x -3 and f(0) = -2

f'(x) = x^2 -5x -3

We need to
find f(x).

We know that f(x) = integral of
f'(x).

==> f(x) = Int ( x^2 -5x -3)
dx

             = Int (x^2) dx - Int (5x) dx - Int 3
dx

              = x^3/3 - 5x^2/2 - 3x +
C

==> f(x)= (1/3)x^3 -(5/2)x^2 - 3x +
C

But we are given that f(0) =
-2

==> f(0) = 0 - 0 - 0 + C =
-2

==> C =
-2

==> f(x) = (1/3)x^3 - (5/2)x^2 - 3x
-2