/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 31 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 31

Evaluate the following integrals. $$\int \frac{\sqrt{x^{2}-9}}{x} d x, x>3$$

Short Answer

Expert verified
Question: Evaluate the integral \(\int \frac{\sqrt{x^{2}-9}}{x} d x\). Solution: Using the trigonometric substitution \(x = 3\sec\theta\), we can simplify the integral to \(\int (3\tan^2 \theta) d\theta\). After evaluating the integral, we get \(3\tan\theta - 3\theta + C = \sqrt{x^2 - 9} - 3\sec^{-1}\frac{x}{3} + C\) as a result.

Step by step solution

01

Choosing the Trigonometric Substitution

To simplify the expression, since we have \(\sqrt{x^2 - 9}\) in the integral, we should choose a trigonometric substitution such that \(x^2 - 9\) becomes a square of a trigonometric function. We can let \(x=3\sec\theta\), with \(\theta\) in \([0,\pi]\). Therefore, derivative with respect to \(\theta\) is: \(\frac{dx}{d\theta} = 3\sec \theta \tan \theta\). Now, we can find an expression for \(\sqrt{x^2 - 9}\) in terms of \(\theta\).
02

Simplifying the Square Root

Using the substitution: $$\sqrt{x^2 - 9} = \sqrt{(3\sec\theta)^2 - 9} = \sqrt{9\sec^2\theta - 9} = \sqrt{9(\sec^2\theta - 1)} = 3\sqrt{\tan^2\theta} = 3\tan\theta$$ Note that we take the positive root since \(x>3\).
03

Changing Variables in the Integral

Replace \(x\) in the integral with our substitution. $$\int \frac{\sqrt{x^{2}-9}}{x} d x = \int \frac{3\tan\theta}{3\sec\theta}\cdot(3\sec \theta \tan \theta) d\theta$$
04

Simplifying the Integral

Cancel out any common terms in the integrand. $$\int \frac{3\tan\theta}{3\sec\theta}\cdot(3\sec \theta \tan \theta) d\theta = \int (3\tan^2 \theta) d\theta$$
05

Evaluating the Integral

Now we can integrate with respect to \(\theta\). $$\int (3\tan^2 \theta) d\theta = 3\int (\sec^2\theta - 1) d\theta = 3\int \sec^2\theta d\theta - 3\int d\theta$$ $$= 3\tan\theta - 3\theta + C$$
06

Back to Original Variable

Now we need to convert our result back in terms of \(x\). To do this, we find the expressions for \(\tan\theta\) and \(\theta\) from our substitution. Recall that \(\tan\theta = \frac{\sqrt{x^2 - 9}}{3}\). $$3\tan\theta - 3\theta + C = 3\left(\frac{\sqrt{x^2 - 9}}{3}\right) - 3\left(\sec^{-1}\frac{x}{3}\right) + C$$ The final answer is: $$\int \frac{\sqrt{x^{2}-9}}{x} d x = \sqrt{x^2 - 9} - 3\sec^{-1}\frac{x}{3} + C$$

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