Chapter 5: Q. 12 (page 494)

Fill in the blanks to complete each of the following theorem statements:
If $a$ is a positive real number, and if $x \in___$ and $u \in_$ , then the substitution $x = a \tan u$ gives us $x^{2} + a^{2} =___$ .

Short Answer

Expert verified

If $a$ is a positive real number, and if $x \in (- \infty, \infty)$ and $u \in (- \frac{π}{2}, \frac{π}{2})$ , then the substitution $x = a \tan u$ gives us $x^{2} + a^{2} = a^{2} s e c^{2} u$ .

Step by step solution

Step 1. Given information

If $a$ is a positive real number, and if $x \in_____$ and $u \in_____$ , then the substitution $x = a \tan u$ gives us $x^{2} + a^{2} =_____$ .

Step 2. Filling in the blanks to complete the theorem statements

If $a$ is a positive real number, and if $x \in (- \infty, \infty)$ and $u \in (- \frac{π}{2}, \frac{π}{2})$ , then the substitution $x = a \tan u$ gives us $x^{2} + a^{2} = a^{2} s e c^{2} u$ .

By substituting for $x$ and simplifying, we get the above answer valid under the conditions mentioned. That is, $x^{2} + a^{2} = {(a \tan u)}^{2} + a^{2}$

$= a^{2} \tan^{2} u + a^{2} = a^{2} (\tan^{2} u + 1) = a^{2} s e c^{2} u$

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Fill in the blanks to complete each of the following theorem statements:
If $a$ is a positive real number, and if $x \in___$ and $u \in_$ , then the substitution $x = a \tan u$ gives us $x^{2} + a^{2} =___$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. Filling in the blanks to complete the theorem statements

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Fill in the blanks to complete each of the following theorem statements: If ais a positive real number, and if x∈_____and u∈_____, then the substitution x=atanugives us x2+a2=_____.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Filling in the blanks to complete the theorem statements

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Probability and Statistics

Discrete Mathematics

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Fill in the blanks to complete each of the following theorem statements:
If $a$ is a positive real number, and if $x \in___$ and $u \in_$ , then the substitution $x = a \tan u$ gives us $x^{2} + a^{2} =___$ .