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

Antidifferentiation of a Composite FunctionChange of VariablesWith a formal

change of variables,you completely rewrite the integral in terms ofuanddu(or any other convenient variable). The change of variable technique uses the Leibniz notation for the differential. That is, itu=g(x), thendu=g'(x)dx, and the integral in Theorem 4.12(shown above) takes the form:ExampleOriginal Function.Write Integral in terms ofu.Constant Multiple Rule.Write the antiderivative in terms ofu.Simplify.Replaceuterms withxterms.Guidelines for Making a Change of Variablesu=g(x). Usually, it is best to choose theinnerpart of a composite function, such as a quantity raised to a power.du=g'(x) dx.u.u.ubyg(x)to obtain an antiderivative in terms ofx.Works Cited: Calculus with Analytic Geometry: Eighth Edition--Larson, Hostetler, and Edwards.Pages used: 295-299Example