/*! 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 34 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 34

Evaluate the following integrals. $$\int \frac{\sqrt{4 x^{2}-1}}{x^{2}} d x, x>\frac{1}{2}$$

Short Answer

Expert verified
Question: Evaluate the integral $$\int \frac{\sqrt{4 x^{2}-1}}{x^{2}} d x, x>\frac{1}{2}$$. Answer: $$\frac{1}{2}\sec^{-1}(2x) + C$$

Step by step solution

01

Perform the trigonometric substitution

Replace \(x\) with \(\frac{1}{2}\sec\theta\): $$\int \frac{\sqrt{4(\frac{1}{2}\sec\theta)^{2}-1}}{(\frac{1}{2}\sec\theta)^{2}} d x$$ Now, we need to find \(dx\) in terms of \(d\theta\). Differentiate \(x = \frac{1}{2}\sec\theta\): $$\frac{dx}{d\theta} = \frac{1}{2}\sec\theta\tan\theta$$ Now, express the integral in terms of \(\theta\).
02

Simplify the integral and substitute \(dx\)

Replace \(dx\) with \(\frac{1}{2}\sec\theta\tan\theta d\theta\): $$\int \frac{\sqrt{4(\frac{1}{2}\sec\theta)^{2}-1}}{(\frac{1}{2}\sec\theta)^{2}} \cdot \frac{1}{2}\sec\theta\tan\theta d\theta$$ Simplify the integral: $$\int \frac{\sqrt{\sec^2\theta-1}}{\sec^2\theta} \cdot \frac{1}{2}\sec\theta\tan\theta d\theta$$ Note that \(\sqrt{\sec^2\theta-1} = \tan\theta\). This allows us to further simplify the integral: $$\int \frac{\tan\theta}{\sec^2\theta} \cdot \frac{1}{2}\sec\theta\tan\theta d\theta$$
03

Integrate with respect to \(\theta\)

Cancel out the \(\tan\theta\) terms and simplify the integral: $$\int \frac{1}{2}d\theta$$ Integrate with respect to \(\theta\): $$\frac{1}{2}\theta + C$$
04

Convert back to \(x\) and find the final solution

Now, we need to convert our result back into \(x\). Since \(x = \frac{1}{2}\sec\theta\), we can use the properties of the secant function to find \(\theta\) in terms of \(x\): $$\sec\theta = 2x \Rightarrow \theta = \sec^{-1}(2x)$$ Replace \(\theta\) with \(\sec^{-1}(2x)\): $$\frac{1}{2}\sec^{-1}(2x) + C$$ There is the final solution: $$\int \frac{\sqrt{4 x^{2}-1}}{x^{2}} d x = \frac{1}{2}\sec^{-1}(2x) + C$$

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Most popular questions from this chapter

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