Integration Summary

Calculus 1 · Axiom Academy

Everything from this unit at a glance — the antiderivative, the Fundamental Theorem, the core techniques, and where it all gets used. Integration is the reverse of differentiation : an antiderivative F satisfies F'(x) = f(x) , and the indefinite integral collects all of them as . The Fundamental Theorem of Calculus ties the two halves of calculus together — differentiation and integration are inverse operations. A definite integral measures net signed area between the curve and the x -axis, and FTC Part 2 evaluates it as F(b) - F(a) . When a direct antiderivative isn't obvious, reach for a technique : u -substitution (reverse chain rule), integration by parts (products), or partial fractions (rational functions). Integration turns a rate into an accumulated total — that's why it computes area, volume, work, and average value. Core Concept The Indefinite Integral Antidifferentiation undoes the derivative. Since any two antiderivatives of f differ by a constant, we tack on the constant of integration C to capture the whole family at once. When to use: recovering a function from its rate of change. Watch out for: dropping the +C — it's part of the answer, not optional. The accumulation function is itself an antiderivative of f : differentiating an integral with a variable upper limit hands you back the integrand. When to use: differentiating a function defined by an integral. Watch out for: a non-trivial upper limit — apply the chain rule (e.g. limit x^2 gives a factor of 2x ).

This is the written version of the interactive lesson above. See the full Calculus 1 course.