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

Identifying When the Substitution Rule is Needed​​

Investigation 05

Identify a potential $\text{d}u$$\text{d}u$. Use this differential to identify an appropriate “inside function” for $u$$u$.

1. ${\displaystyle \int \frac{e^x}{1+e^{2x}}\text{d}x}$$\displaystyle \int{\frac{{e^x}}{1+e^{2x}}\,{\text{d}x}}$​

2. ${\displaystyle \int \frac{x}{1+x^4}\text{d}x}$$\displaystyle \int{\frac{x}{1+x^4}\,\text{d}x}$

3. ${\displaystyle \int \frac{\sin (x)}{\sqrt{1-{\cos }^4(x)}}\text{d}x}$$\displaystyle \int{\frac{\sin{(x)}}{\sqrt{1-\cos^4{\!(x)}}}\,{\text{d}x}}$

4. ${\displaystyle \int \frac{1}{x\ln (x)}\text{d}x}$$\displaystyle \int{\frac{1}{x\,\ln{(x)}}\,\text{d}x}$

5. ${\displaystyle \int x^2{(x^6+5)}^2\text{d}x}$$\displaystyle \int{x^2\,\left(x^6+5\right)^{2}\,{\text{d}x}}$

6. ${\displaystyle \int \frac{e^x+1}{e^x+x}\text{d}x}$$\displaystyle \int{\frac{e^x+1}{e^x+x}\,\text{d}x}$

7. ${\displaystyle \int \frac{{\sec }^2(x)}{\tan (x)\sqrt{{\tan }^2(x)-1}}\text{d}x}$$\displaystyle \int{\frac{\sec^2{\!(x)}}{\tan{(x)}\sqrt{\tan^2{\!(x)}-1}}\,{\text{d}x}}$

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Created: Monday, 15 August 2022 - 17:12 (EDT)