Integration Strategy

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

• Section 7.5 – Strategy for Integration

Expected Educational Results

• Objective 08–01: I can identify the appropriate method to evaluate an 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.

Practice

• Identify the method(s) needed to evaluate the following integrals. Explain.

• Identify $u$$u$, $du$$du$, $U$$U$, $dV$$dV$, where needed. Explain.

• Evaluate the integral.

In each of the following sets of integrals, the integrals are slightly different from each other. This requires a different method to evaluate each integral. This will help you determine the valid method for integration.

Investigation 01

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

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

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

4. ${\displaystyle \int \frac{x^3}{1-x^2}dx}$$\displaystyle \int{\frac{x^3}{1-x^2}\,dx}$

Investigation 02

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

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

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

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

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

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

Investigation 03

1. ${\displaystyle \int {\sin }^3(x){\cos }^5(x)dx}$$\displaystyle \int{\sin^3{(x)}\cos^5{(x)}\,dx}$

2. ${\displaystyle \int \sin (x){\cos }^2(x)dx}$$\displaystyle \int{\sin{(x)}\cos^2{(x)}\,dx}$

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

4. ${\displaystyle \int {\sin }^2(x){\cos }^2(x)dx}$$\displaystyle \int{\sin^2{(x)}\cos^2{(x)}\,dx}$

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

Investigation 04

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

2. ${\displaystyle \int {\tan }^4(x){\sec }^2(x)dx}$$\displaystyle \int{\tan^4{(x)}\sec^2{(x)}\,dx}$

3. ${\displaystyle \int {\tan }^4(x){\sec }^4(x)dx}$$\displaystyle \int{\tan^4{(x)}\sec^4{(x)}\,dx}$

4. ${\displaystyle \int \tan (x)dx}$$\displaystyle \int{\tan{(x)}\,dx}$

5. ${\displaystyle \int {\tan }^2(x)dx}$$\displaystyle \int{\tan^2{(x)}\,dx}$

6. ${\displaystyle \int {\sec }^2(x)dx}$$\displaystyle \int{\sec^2{(x)}\,dx}$

Investigation 05

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

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

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

Investigation 06

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

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

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

Investigation 07

1. ${\displaystyle \int \frac{1}{x\sqrt{\ln (x)}}dx}$$\displaystyle \int{\frac{1}{x\sqrt{\ln{(x)}}}\,dx}$

2. ${\displaystyle \int \frac{\ln (x)}{x}dx}$$\displaystyle \int{\frac{\ln{(x)}}{x}\,dx}$

3. ${\displaystyle \int x\ln (x)dx}$$\displaystyle \int{x\ln{(x)}\,dx}$

4. ${\displaystyle \int \ln (x)dx}$$\displaystyle \int{\ln{(x)}\,dx}$

Investigation 08

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

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

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

Investigation 09

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

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

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

4. ${\displaystyle \int {\sin }^{-1}(x)dx}$$\displaystyle \int{\sin^{-1}{(x)}\,dx}$

Investigation 10

1. ${\displaystyle \int \frac{e^{\sqrt{x}}}{\sqrt{x}}dx}$$\displaystyle \int{\frac{e^{\sqrt{x}}}{\sqrt{x}}\,dx}$

2. ${\displaystyle \int \sqrt{x}{e}^{\sqrt{x}}dx}$$\displaystyle \int{\sqrt{x}e^{\sqrt{x}}\,dx}$

3. ${\displaystyle \int e^{\sqrt{x}}dx}$$\displaystyle \int{e^{\sqrt{x}}\,dx}$

Investigation 11

1. ${\displaystyle \int \frac{{\sec }^2(x)}{1+\tan (x)}dx}$$\displaystyle \int{\frac{\sec^2{(x)}}{1+\tan{(x)}}\,dx}$

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

Investigation 12

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

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

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

4. ${\displaystyle \int \frac{1}{x^2\sqrt{x^2-4}}dx}$$\displaystyle \int{\frac{1}{x^2\sqrt{x^2-4}}\,dx}$

Investigation 13

1. ${\displaystyle \int (3x^2+x-5)e^{-x}dx}$$\displaystyle \int{(3x^2+x-5)e^{-x}\,dx}$

2. ${\displaystyle \int e^{3x}\cos (5x)dx}$$\displaystyle \int{e^{3x}\cos{(5x)}\,dx}$

Investigation 14

1. ${\displaystyle \int \frac{1}{e^{-x}+e^x}dx}$$\displaystyle \int{\frac{1}{e^{-x}+e^x}\,dx}$

Investigation 15

1. ${\displaystyle \int \frac{2x^3+15x^2+21x-10}{\sqrt{x^2+5x-2}}dx}$$\displaystyle \int{\frac{2x^3+15x^2+21x-10}{\sqrt{x^2+5x-2}}\,dx}$

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Created: Monday, 6 September 2020 13:33 EDT Last Modified: Sunday, 13 February 2022 - 09:48 (EST)