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 03

What pattern(s) do you notice about the following indefinite integrals?

1. ${\displaystyle \int 2x\cos \left(x^2\right)\text{d}x}$$\displaystyle \int{{2x}\,\cos{\left({x^2}\right)}\,{\text{d}x}}$

2. ${\displaystyle \int \frac{{\left[\ln (x)\right]}^3}{x}\text{d}x}$$\displaystyle \int{\frac{\left[{\ln{(x)}}\right]^3}{{x}}\,{\text{d}x}}$

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

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

5. ${\displaystyle \int \frac{x}{\sqrt{1-x^2}}\text{d}x}$$\displaystyle \int{\frac{x}{\sqrt{1-x^2}}\,\text{d}x}$

6. ${\displaystyle \int \frac{1}{3x+2}\text{d}x}$$\displaystyle \int{\frac{1}{3x+2}\,\text{d}x}$

7. ${\displaystyle \int 3x^2{e}^{x^3}\text{d}x}$$\displaystyle \int{3x^2\,e^{x^3}\,{\text{d}x}}$

8. ${\displaystyle \int e^x\cos \left(e^x\right)\text{d}x}$$\displaystyle \int{e^x\,\cos{\left(e^x\right)}\,{\text{d}x}}$

9. ${\displaystyle \int \cos \left(x\right)e^{\sin (x)}\text{d}x}$$\displaystyle \int{\cos{\left(x\right)\,e^{\sin{(x)}}}\,{\text{d}x}}$

10. ${\displaystyle \int 4x^3{(x^4+5)}^{10}\text{d}x}$$\displaystyle \int{4x^3\,\left(x^4+5\right)^{10}\,{\text{d}x}}$

11. ${\displaystyle \int {\sin }^4(x)\cos (x)\text{d}x}$$\displaystyle \int{{\sin}^4{(x)}\,{\cos{(x)}}\,{\text{d}x}}$

12. ${\displaystyle \int \sin (x){\cos }^2(x)\text{d}x}$$\displaystyle \int{{\sin}{(x)}\,{\cos^2{\!(x)}}\,{\text{d}x}}$

13. ${\displaystyle \int \cos (x)\cos (\sin (x))\text{d}x}$$\displaystyle \int{\cos{(x)}\,\cos{\left(\sin{(x)}\right)}\,{\text{d}x}}$

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Created: Monday, 9 August 2021 18:16 EDT Last Modified: Monday, 15 August 2022 - 16:08 (EDT)