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Understanding U-Substitution in Antiderivatives of Composite Functions

This section introduces the concept of finding antiderivatives using u-substitution, a technique crucial for solving integrals involving composite functions. By letting u = g(x), where g is a differentiable function, we find its differential du = g'(x)dx. The method emphasizes pattern recognition, allowing students to manipulate integrals effectively. Examples illustrate how to rewrite integrals using u-substitution, such as substituting expressions for dx and u to simplify calculations. This foundational knowledge is vital for mastering calculus.

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Understanding U-Substitution in Antiderivatives of Composite Functions

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  1. Section 4.5 Notes 1st Day HW: p.304 (11-37 odd, 43-61odd)

  2. Finding the Antiderivative of a Composite Function

  3. or • Let u = g(x), then du = gꞌ(x) dx and • The last form is called u-substitution. One technique of using u-substitution is called pattern recognition.

  4. Examples

  5. Notice that 2 is the derivative of 1 + 2x. • Let u = 1 + 2x now take the derivative of both sides of this expression. • du = 2 dx • Now substitute u in for 1 + 2x and dufor (2) dx in the integral expression. • Integrate this expression using the rules from p. 250.

  6. Now substitute 1 + 2x in for u. • This is the pattern recognition technique.

  7. u = 3x – 1 • du = 3 dx • There is no 3 dx in the expression only a dx. • Substitute this expression in for dx.

  8. u = x3 + 5 • du = 3x2dx • Rewrite the integral to use u-substitution.

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