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Chapter 5: Q. 0 (page 463)

Problem Zero: Read the section and make your own summary of the material.

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Step 1. Given information.

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Read the section and make your own summary of the material.

02

Step 2. Read the section and make your own summary of the material. 

Three Types of Trigonometric Substitutions

Suppose a is a positive real number. The following substitutions can be useful in solving certain integrals:

(a) If x=asin⁡u, thena2−x2=acos⁡u.

This substitution is valid for x∈[−a,a]andu∈−π2,π2 .

(b) If x=atan⁡u,thenx2+a2=a2sec2⁡u.

This substitution is valid for x∈(−∞,∞)andu∈−π2,π2 .

(c) If x=asec⁡u,thenx2−a2=atan⁡u,ifx>a−atan⁡u,ifx<−a

This substitution is valid for x∈(−∞,−a]∪[a,∞)andu∈0,π2∪π2,π.

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Most popular questions from this chapter

Solve the integral

∫1x2x2-9dx

Complete the square for each quadratic in Exercises 28–33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

x2-5x+1

Solve the integral

∫3-xx-1dx

Find three integrals in Exercises 27–70 for which a good strategy is to use integration by parts with u=xand dv the remaining part.

Solve given integrals by using polynomial long division to rewrite the integrand. This is one way that you can sometimes avoid using trigonometric substitution; moreover, sometimes it works when trigonometric substitution does not apply.

∫x4-32+3x2dx
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