Integration as Anti-Differentiation

Integration is the reverse process of differentiation. If differentiation takes a function and finds its rate of change, integration goes the other direction — given a rate of change, it recovers the original function. For this reason, integration is often called anti-differentiation.

Anti-derivative

A function $F (x)$ is called an anti-derivative of $f (x)$ if $F^{'} (x) = f (x)$ . In other words, differentiating $F (x)$ returns $f (x)$ .

For example, since $\frac{d}{d x} (x^{3}) = 3 x^{2}$ , we know that $x^{3}$ is an anti-derivative of $3 x^{2}$ .

Note

The anti-derivative is not unique. Because the derivative of any constant is zero, both $x^{3} + 5$ and $x^{3} - 7$ are equally valid anti-derivatives of $3 x^{2}$ . This is why we always include a constant of integration, $C$ , when writing an indefinite integral.

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SL 5.5—Integration introduction, areas between curve and x axis

Integration as Anti-Differentiation

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