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Section 4.5 – Integration by Substitution. The Chain Rule and Integration. du. u. Find if . THUS:. Integration by Substitution. Let f , g , and u be differentiable functions of x such that Then where G is an antiderivative of g .

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The Chain Rule and Integration du u Find if . THUS:

Integration by Substitution Let f, g, and u be differentiable functions of x such that Then where G is an antiderivative of g.

Substitution Guidelines • 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. • Compute du = g ‘(x)dx. • Rewrite the integral in terms of u. Make sure every x and dx is no longer in the integral. • Find the resulting integral in terms of u. • Replace uby g(x)to obtain an antiderivative in terms of x. • If you have a definite integral, make sure you change the limits of integration to be in terms of ubefore you integrate.

Example 1 Define u and du: Substitute to replace EVERY x and dx: Integrate. Substitute back to Leave your answer in terms of x. Find .

Example 2 Define u and du: Substitute to replace EVERY x and dx: Solve for dx Integrate. Substitute back to Leave your answer in terms of x. Find .

Example 3 Rewrite Define u and du: Substitute to replace EVERY x and dx: Solve for dx Integrate.` Substitute back to Leave your answer in terms of x. Find

Example 4 Define u and du: Solve for dx Substitute to replace EVERY x and dx: There is still an x. Solve the initial equation of u for x. Integrate. Substitute back to Leave your answer in terms of x. Find .

Example 5 Define u and du: Substitute to replace EVERY x and dx: Solve for dx Find C: Find F(x) if F(0.5) = 4 and .

Example 6 Change the Limits: Define u and du: Substitute: Path 1: Path 2: Evaluate

Example 7 Change the Limits: Define u and du: Substitute: Path 1: Path 2: Evaluate

White Board Challenge Evaluate:

Integration of Even and Odd Functions Let f be integrable on the interval [-a,a]. • If f is an even function, then -a a

Integration of Even and Odd Functions Let f be integrable on the interval [-a,a]. • If f is an odd function, then -a a