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Calculus I / The Integral

Indefinite Integrals, the Net Change Theorem, and the Substitution Rule

The FTC Part II reduces definite integrals to antiderivatives, but the antidifferentiation table in Antiderivatives and Initial-Value Problems only handles simple powers, exponentials, and basic trig. Most real-world integrals involve composite functions — a function of a function — for which the table is silent. The substitution rule (also called uu-substitution) is the chain rule read right-to-left: it handles the integral of any function of the form f(g(x))⋅g′(x) dxf(g(x))\cdot g'(x)\,dx by substituting u=g(x)u = g(x), turning the integral into ∫f(u) du\int f(u)\,du, which the table can then handle.
Before introducing substitution, this note formalises the indefinite integral notation and states the Net Change Theorem — FTC Part II rephrased as "the integral of a rate of change equals the total change" — which is the form most useful in financial applications.
Learning objectives. By the end of this note you should be able to:
  • state and apply the Net Change Theorem: ∫abF′(x) dx=F(b)−F(a)\int_a^b F'(x)\,dx = F(b) - F(a);
  • identify when a substitution u=g(x)u = g(x) simplifies an integral;
  • execute uu-substitution for indefinite integrals, remembering to back-substitute for xx;
  • execute uu-substitution for definite integrals by changing the limits appropriately;
  • recognise when an integrand is symmetric (even/odd) and exploit it.

In this lesson

  1. Indefinite integrals: a uniform notation
  2. The Net Change Theorem
  3. The Substitution Rule
  4. Substitution in definite integrals
  5. Symmetry and definite integrals

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