6.2 – Antidifferentiation by Substitution

Introduction Our antidifferentiation formulas don’t tell us how to evaluate integrals such as Our strategy is to change from the variable x to a new variable u. What should we let u = ? Du??

Example 1 Evaluate

The Substitution Rule In general this method works whenever we have an integral that we can write in the form We make the “change of variable” or “substitution” u = g(x): The Substitution Rule If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then

Example 2 Evaluate

Remarks The main challenge in using the Rule is to think of an appropriate substitution. You should try to choose u to be… Some function in the integrand whose differential also occurs (except for a constant) …or some complicated part of the integrand (perhaps the inner function in a composite function).

Example 3 Evaluate

Example 4 Evaluate

Example 5 Calculate

Example 6 Calculate

Definite Integrals When evaluating a definite integral by substitution, two methods are possible. One method is to evaluate the indefinite integral first and then use the Evaluation Theorem, for example

Definite Integrals (cont’d) Another method is to change the limits of integration when the variable is changed: The Substitution Rule for Definite Integrals If g’ is continuous on [a,b] and f is continuous on the range of u = g(x), then

Example 7 Evaluate again, this time changing the limits of integration

Example 8 Evaluate

Example 9 Evaluate

Symmetry We can use the Substitution Rule for Definite Integrals to simplify the calculation of integrals of functions that possess symmetry properties: Integrals of Symmetric Functions Suppose f is continuous on [-a,a]. (a) If f is even [f(-x) = f(x)], then (b) If f is odd [f(-x) = -f(x)], then

Example 10 Evaluate

Example 11 Evaluate