Anti-differentiation and the Indefinite Integral

When taking the anti-derivative of a polynomial, the result is a set of functions that differ by a constant. We call this set the general anti-derivative.

Here is a graph of x² (green) and it’s anti-derivative x³/3 + C (red). We can think of the integrand x² as the curve that defines the slope of the anti-derivative x³/3 at every point along the x axis. Note that where the slope of x³/3 is positive, the value of x² is positive and where the slope of x³/3 is zero, the value of x² is zero.

The value of the integrand x² only tells us about the slope of the anti-derivative x³/3. It doesn’t contain any information about whether the anti-derivative is translated, up or down, along the y axis. As an example, x³/3+1 and x³/3−1 are equally valid anti-derivatives of x². We can indicate this fact by adding an arbitrary constant C to the anti-derivative resulting in x³/3+C. We do this and we say that the resulting anti-derivative expresses a set of functions that differ by a constant. We call this set the general anti-derivative.

We can resolve the general anti-derivative to a specific function if we have an initial condition which essentially gives us the value of the arbitrary constant C or we can get a numerical answer for the anti-derivative if we have limits of integration on the integral. In the second case, the…