From antiderivative to definite integral
Find an antiderivative and see why +C is needed, evaluate a definite integral with F(b) − F(a), find the mean value of a function and meet the integral of 1x.
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Guide
Integrating is the reverse of differentiating. An antiderivative of f is a function F with F′(x) = f(x). Shifting a graph up or down does not change its gradient, so every antiderivative comes with a constant: ∫f(x)dx = F(x) + C (the indefinite integral).
For the definite integral from a to b, find F and calculate F(b) − F(a): the upper end first, then take away the lower end. It is the area under the graph (below the axis it counts as negative).
The mean value of f on [a, b] is the definite integral divided by the width b − a. The power rule ∫xndx = xn + 1/(n + 1) + C works for every n except −1: ∫1xdx = ln|x| + C.