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Chapter 7: Problem 2

What change of variables is suggested by an integral containing \(\sqrt{x^{2}+36} ?\)

Short Answer

Expert verified
Answer: The change of variables suggested is \(x = 6\tan(\theta)\), which transforms the given expression into \(6\sec(\theta) ~ 6\sec^2(\theta)d\theta\).

Step by step solution

01

Identify the appropriate trigonometric substitution

\(x = 6\tan(\theta)\), which means \(\theta = \arctan(\frac{x}{6})\). This allows us to rewrite the given expression using the Pythagorean trigonometric identity.
02

Calculate the differential of x in terms of 胃

Using the substitution from Step 1, find the differential \(dx\): \(dx = \frac{d(6\tan(\theta))}{d\theta} d\theta = 6\sec^2(\theta)d\theta\)
03

Substitute x and dx into the given expression

Replace \(x\) with the substitution found in Step 1 and \(dx\) with the result from Step 2 in the given expression, which is \(\sqrt{x^2 + 36}\): \(\sqrt{(6\tan(\theta))^2 + 36} ~ 6\sec^2(\theta)d\theta\)
04

Simplify the expression

Now, simplify the expression and observe the trigonometric identity: \(\sqrt{36\tan^2(\theta) + 36} ~ 6\sec^2(\theta)d\theta\) \(\sqrt{36(\tan^2(\theta) + 1)} ~ 6\sec^2(\theta)d\theta\) \(6\sqrt{\sec^2(\theta)} ~ 6\sec^2(\theta)d\theta\) \(6\sec(\theta) ~ 6\sec^2(\theta)d\theta\)
05

Conclusion

The change of variables suggested by the integral containing \(\sqrt{x^2 + 36}\) is \(x = 6\tan(\theta)\), which transforms the given expression into \(6\sec(\theta) ~ 6\sec^2(\theta)d\theta\).

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