Chapter 5: Q. 14 (page 464)

Explain why, if $x = a \tan u$ , then $\sqrt{x^{2} + a^{2}} = a s e c u$ . Your explanation should include a discussion of domains and absolute values.

Short Answer

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Ans:

$\sqrt{x^{2} + a^{2}} = a s e c u . . . . . . . [x = a \tan u] = \sqrt{a^{2} \tan^{2} u + a^{2}} = a s e c u$

Step by step solution

Step 1. Given information:

$x = a \tan u \sqrt{x^{2} + a^{2}} = a s e c u$

Step 2. Solving the trigonometric substitution:

If $x = a \tan u$ , there is no domain restriction on $x$ , because $x = a \tan u$ can be any real number. This means that trigonometric substitution $x = a \tan u x$ can be used even for integrands that do not have square root.

The domain of $x^{2} + a^{2}$ can be any value.

So, the integral involving $x^{2} + a^{2}$ are well suited for trigonometric substitution with $x = a \tan u$ .

Using the Pythagorean identity $a^{2} \tan^{2} u + a^{2} = a^{2} \sec^{2} u$ .

$\sqrt{x^{2} + a^{2}} = \sqrt{a^{2} \tan^{2} u + a^{2}} = \sqrt{a^{2} s e c^{2} u} = |a s e c u| [A . \sqrt{A^{2}} = |A|] = a s e c u$

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Explain why, if $x = a \tan u$ , then $\sqrt{x^{2} + a^{2}} = a s e c u$ . Your explanation should include a discussion of domains and absolute values.

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Solving the trigonometric substitution:

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Explain why, if x=atanu, then x2+a2=asecu. Your explanation should include a discussion of domains and absolute values.

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Solving the trigonometric substitution:

One App. One Place for Learning.

Most popular questions from this chapter

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Explain why, if $x = a \tan u$ , then $\sqrt{x^{2} + a^{2}} = a s e c u$ . Your explanation should include a discussion of domains and absolute values.