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Chapter 7: Problem 12

Use a table of integrals to determine the following indefinite integrals. $$\int \frac{d x}{x \sqrt{81-x^{2}}}$$

Short Answer

Expert verified
Answer: \(\ln\left|\frac{9}{x} + \frac{\sqrt{81 - x^2}}{x}\right| + C\)

Step by step solution

01

Trigonometric Substitution

In this integral, the expression \(81 - x^2\) inside the square root suggests the use of a trigonometric substitution. Since \(\sin^2{\theta} + \cos^2{\theta} = 1\), we can transform \(81 - x^2\) into a simpler form by letting \(x = 9\sin{\theta}\) and \(dx = 9\cos{\theta}d\theta\). The relationship \(\sin^2{\theta} + \cos^2{\theta} = 1\) implies that \(\frac{81 - x^2}{81} = \cos^2{\theta}\).
02

Rewrite the Integral

Substitute \(x = 9\sin{\theta}\) and \(dx = 9\cos{\theta}d\theta\) into the integral: $$\int \frac{d x}{x \sqrt{81-x^{2}}} = \int \frac{9\cos{\theta}d\theta}{(9\sin{\theta}) \sqrt{81(1 - \sin^2{\theta})}}$$ Simplify the integral: $$\int \frac{9\cos{\theta}d\theta}{(9\sin{\theta}) \cdot 9\cos{\theta}} = \int \frac{1}{\sin{\theta}}d\theta$$ Now, we have a simpler integral to solve using a table of integrals.
03

Solve the Integral Using a Table of Integrals

Using a table of integrals or known results, we can find that the indefinite integral of \(\frac{1}{\sin{\theta}}\) is given by: $$\int \frac{1}{\sin{\theta}}d\theta = \ln\left|\csc{\theta} + \cot{\theta}\right| + C$$ where \(C\) is the constant of integration.
04

Convert Back to the Original Variable

Now we need to convert back to the original variable \(x\). Recall that we initially made the substitution \(x = 9\sin{\theta}\). Solving this equation for \(\theta\), we have \(\theta = \arcsin{\frac{x}{9}}\). Moreover, we also have: $$\cos{\theta} = \sqrt{1 - \sin^2{\theta}} = \frac{\sqrt{81 - x^2}}{9}$$ $$\cot{\theta} = \frac{\cos{\theta}}{\sin{\theta}} = \frac{\sqrt{81 - x^2}}{x}$$ $$\csc{\theta} = \frac{1}{\sin{\theta}} = \frac{9}{x}$$ Substitute these expressions back into the integral: $$\int \frac{1}{x \sqrt{81-x^{2}}} dx = \ln\left|\frac{9}{x} + \frac{\sqrt{81 - x^2}}{x}\right| + C$$ This is the final result for the indefinite integral.

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