Substitution Rule

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 5.5 – The Substitution Rule

Expected Educational Results

• Objective 02–01: I can use a substitution to evaluate an indefinite integral using FTC-I.

• Objective 02–02: I can use a substitution to evaluate a definite integral.

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Substitution Rule

Definition: Substitution Rule for Definite Integrals

If $u=g(x)$$u=g(x)$ is a differentiable function, then

${\int }_a^{b}f(g(x))g^{\prime }(x)dx={\int }_{g(a)}^{g(b)}f(u)du$$\int_a^b{f\left(g(x)\right)g^{\,\prime}(x)\,dx}=\int_{g(a)}^{g(b)}{f(u)\,du}$​

Investigation 02

1. What do you notice about the integrand on the left side of the equation in the definition above? Explain.

2. What do you notice about the integrand on the right side of the equation in the definition above? Explain.

Check your Understanding

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Last Modified: Monday, 9 August 2021 17:13 EDT