Integrals

This branch moves from antiderivatives and signed area to the fundamental theorem of calculus, common identities, substitution-style techniques, integration by parts, improper integrals, and numerical approximation.

Use the integrals cheat sheet for quick lookup after the concepts are clear.

Reading route and scope

This is a reference for real, one-variable integration, not measure theory or multivariable integration. Start with limits, continuity, and the derivative’s product and chain rules.

1. Definitions: distinguish a number accumulated over an interval from a family of antiderivatives.
2. Fundamental theorem: connect the two under continuity assumptions.
3. Properties and formulas: check domains before using a lookup.
4. Substitution and integration by parts: reverse the chain and product rules; the latter also covers trigonometric substitution and partial fractions.
5. Improper and numerical integrals: decide whether a value exists before approximating it, then distinguish quadrature error from tail error.

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