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Chapter 7: Problem 19

Use a table of integrals to determine the following indefinite integrals. $$\int \frac{d x}{x \sqrt{144-x^{2}}}$$

Short Answer

Expert verified
Question: Evaluate the indefinite integral: $$\int \frac{dx}{x\sqrt{144-x^2}}$$ Answer: The indefinite integral is $$\frac{1}{12}\csc^{-1}\left(\frac{x}{12}\right) + C$$.

Step by step solution

01

Identify the integral formula

First, identify the integral formula from a table of integrals that matches the given integrand. The formula needed is: $$\int \frac{du}{u\sqrt{a^2-u^2}} = \frac{1}{a}\csc^{-1}\left (\frac{u}{a}\right )+C$$ Here, our goal is to rewrite our given integral to match this formula.
02

Substitute variables in the integral

Write down the given integral, and compare it to the general formula, identifying the respective values for \(a\) and \(u\). Given integral: $$\int \frac{dx}{x\sqrt{144-x^2}}$$ From the formula: $$\int \frac{du}{u\sqrt{a^2-u^2}} = \frac{1}{a}\csc^{-1}\left (\frac{u}{a}\right )+C$$ Then, \(u = x\) and \(a^2 = 144 \Rightarrow a = 12\)
03

Apply the integral formula

Now, we can apply the formula to compute the indefinite integral. $$\int \frac{dx}{x\sqrt{144-x^2}} = \frac{1}{12}\csc^{-1}\left(\frac{x}{12}\right) + C$$ The result is: $$\int \frac{dx}{x\sqrt{144-x^2}} = \frac{1}{12}\csc^{-1}\left(\frac{x}{12}\right) + C$$

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