Problem

Graph sin²(x), then use the graph to sketch an antiderivative of sin²(x).

Solution

Here is sin²(x).

This graph gives the slope of the graph of its antiderivative at each point. Since the graph of y = sin²(x) is never negative, the graph of the antiderivative must be increasing or at the least nondecreasing at every point. The slope of a line tangent to the graph of the antiderivative will be steepest when the value of sin²(x) is greatest, at p/2, 3p/2, and so on.

Keeping all this in mind, here is a sketch of the graph, with the arbitrary constant C chosen so that the graph passes through the origin: