Mr. Integral
Math notes/Calculus I/The Integral

Definite Integrals and the Fundamental Theorem — Cheat sheet

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Core rule

∫abf=F(b)−F(a),\int_a^bf=F(b)-F(a), ddx∫ag(x)f(t)dt=f(g(x))g′(x).\frac d{dx}\int_a^{g(x)}f(t)dt=f(g(x))g'(x).

For continuous ff, average value is 1b−a∫abf\frac1{b-a}\int_a^bf.

Watch for

Check continuity or justified integrability and the antiderivative hypotheses. Keep variable-bound chain factors; never skip an interior singularity.