Tables of Integrals

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

• Section 7.6 – Integration Using Tables and Computer Algebra Systems

Expected Educational Results

• Objective 07–01: I can use a Table of Integrals to evaluate an indefinite integral.

• Objective 07–02: I can use an appropriate Substitution, then use a Table of Integrals to evaluate an indefinite integral.

• Objective 07–03: I can use a Table of Integrals to evaluate a definite integral using FTC-II.

• Objective 07–04: I can use an appropriate Substitution, then use a Table of Integrals to evaluate a definite integral using FTC-II.

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.

Using a Table of Integrals

NOTE: Table of Integrals are used either

1. to quickly integrate functions when we already know how to integrate the function; or

2. when the method for integration uses methods beyond our understanding.

Steps to using Tables of Integrals:

1. Determine if a previously-discussed method for evaluating integrals is valid for the integral.

2. Simplify the integrand, if possible.

3. Classify the integrand according to its form.

4. Identify the particular integral form.

5. Apply the antiderivative as shown in the integral form.

Investigation 02

Open TABLE_07_Table_Of_Integrals.pdf.

Use the Table of Integrals to integrate the following:

1. ${\displaystyle \int \frac{1}{x\sqrt{2x+3}}dx}$$\displaystyle \int{\frac{1}{x\sqrt{2x+3}}\,dx}$

2. ${\displaystyle \int \frac{1}{x^2\sqrt{3+x^2}}dx}$$\displaystyle \int{\frac{1}{x^2\sqrt{3+x^2}}\,dx}$

3. ${\displaystyle \int \frac{\sqrt{4-x^2}}{x}dx}$$\displaystyle \int{\frac{\sqrt{4-x^2}}{x}\,dx}$

4. ${\displaystyle \int \frac{x^2}{\sqrt{6x-x^2}}dx}$$\displaystyle \int{\frac{x^2}{\sqrt{6x-x^2}}\,dx}$

5. ${\displaystyle \int \frac{1}{{(x^2-5)}^{\frac{3}{2}}}dx}$$\displaystyle \int{\frac{1}{\left(x^2-5\right)^\frac{3}{2}}\,dx}$

6. ${\displaystyle \int e^{3x}\cos (2x)dx}$$\displaystyle \int{e^{3x}\cos{(2x)}\,dx}$

7. ${\displaystyle \int {(\ln (x))}^{2}dx}$$\displaystyle \int{\left(\ln{(x)}\right)^2\,dx}$

8. ${\displaystyle \int {\sec }^3(x)dx}$$\displaystyle \int{\sec^3{(x)}\,dx}$

9. ${\displaystyle \int \frac{1}{x\sqrt{8x-x^2}}dx}$$\displaystyle \int{\frac{1}{x\sqrt{8x-x^2}}\,dx}$

10. ${\displaystyle \int x^2{\cos }^{-1}(x)dx}$$\displaystyle \int{x^2\cos^{-1}{(x)}\,dx}$

11. ${\displaystyle \int \sin (5x)\cos (4x)dx}$$\displaystyle \int{\sin{(5x)}\cos{(4x)}\,dx}$

12. ${\displaystyle \int \frac{\sqrt{5x+7}}{x^2}dx}$$\displaystyle \int{\frac{\sqrt{5x+7}}{x^2}\,dx}$

13. ${\displaystyle \int {\cos }^3(x)dx}$$\displaystyle \int{\cos^3{(x)}\,dx}$

14. ${\displaystyle \int \frac{1}{x^2(7-3x)}dx}$$\displaystyle \int{\frac{1}{x^2(7-3x)}\,dx}$

15. ${\displaystyle \int \frac{2}{(7x-1{)}^2}dx}$$\displaystyle \int{\frac{2}{(7x-1)^2}\,dx}$

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Created: Sunday, 12 September 2021 - 18:15 (EDT) Last Modified: Friday, 10 June 2022 - 10:59 (EDT)