Antidifferentiation of a Composite Function


Change of Variables
With a formal change of variables, you completely rewrite the integral in terms of u and du (or any other convenient variable). The change of variable technique uses the Leibniz notation for the differential. That is, it u=g(x), then du=g'(x)dx, and the integral in Theorem 4.12(shown above) takes the form:

Example

    • Original Function.

    • Write Integral in terms of u.

    • Constant Multiple Rule.

    • Write the antiderivative in terms of u.

    • Simplify.

    • Replace u terms with x terms.

Guidelines for Making a Change of Variables
  • 1. Choose a substitution u=g(x). Usually, it is best to choose the inner part of a composite function, such as a quantity raised to a power.
  • 2. Compute du=g'(x) dx.
  • 3. Rewrite the integral in terms of the variable u.
  • 4. Find the resulting integral in terms of u.
  • 5. Replace u by g(x) to obtain an antiderivative in terms of x.
  • 6. Check your answer by differentiating.

Works Cited: Calculus with Analytic Geometry: Eighth Edition--Larson, Hostetler, and Edwards.
Pages used: 295-299


Example