/*! 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 40 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 40

Evaluate the following integrals. $$\int \frac{d x}{x^{2} \sqrt{9 x^{2}-1}}, x>\frac{1}{3}$$

Short Answer

Expert verified
The result of the integral is given by: $$\int \frac{d x}{x^{2} \sqrt{9 x^{2}-1}} = \sqrt{3}\sec^{-1}(3x) + C$$

Step by step solution

01

Perform the trigonometric substitution

To simplify the integrand, let's first perform the substitution: $$x = \frac{1}{3} \sec\theta$$ Differentiating both sides with respect to \(\theta\), we get: $$dx = \frac{1}{3} \sec\theta \tan\theta d\theta$$
02

Replace \(x\) and \(dx\) in the integral

Now, we replace \(x\) and \(dx\) in the integral using the substitution: $$\int \frac{d x}{x^{2} \sqrt{9 x^{2}-1}} = \int \frac{\frac{1}{3} \sec\theta \tan\theta d\theta}{(\frac{1}{3}\sec\theta)^2 \sqrt{9(\frac{1}{3}\sec\theta)^2 -1}}$$
03

Simplify the integrand

Now we simplify the integrand: $$\int \frac{\frac{1}{3} \sec\theta \tan\theta d\theta}{(\frac{1}{9}\sec^2\theta) \sqrt{3\sec^2\theta -1}}$$ Combine the constants and cancel out common terms: $$\int \frac{(3) \sec\theta \tan\theta d\theta}{\sec^2\theta \sqrt{3\sec^2\theta -1}}$$
04

Simplify further using trigonometric identities

We notice that the integral can be simplified using the trigonometric identity: \(1 + \tan^2\theta = \sec^2\theta\): $$\int \frac{3 \tan\theta \sec\theta d\theta}{\sec^2\theta \sqrt{3(\sec^2\theta -1)}}$$ Cancel out \(\sec^2\theta\): $$\int \frac{3 \tan\theta d\theta}{\sqrt{3(\sec^2\theta -1)}}$$ Now, substitute \(\sec^2\theta - 1 = \tan^2\theta\) back: $$\int \frac{3 \tan\theta d\theta}{\sqrt{3\tan^2\theta}}$$
05

Simplify and integrate

Simplify the integrand one last time, then integrate: $$\int \frac{3 \tan\theta d\theta}{\sqrt{3}\tan\theta}$$ Cancel out \(\tan\theta\): $$\int \frac{3 d\theta}{\sqrt{3}}$$ Now, integrate the simplified integrand: $$\frac{3}{\sqrt{3}}\int d\theta = \frac{3}{\sqrt{3}}\theta + C$$
06

Substitute back in terms of \(x\)

Now, we need to express our result in terms of \(x\). Recall the initial substitution: \(x = \frac{1}{3} \sec\theta\). We can find \(\theta\) as follows: $$\theta = \sec^{-1}(3x)$$ Substituting \(\theta\) back into the result: $$\frac{3}{\sqrt{3}}\sec^{-1}(3x) + C = \sqrt{3}\sec^{-1}(3x) + C$$ So the final result is: $$\int \frac{d x}{x^{2} \sqrt{9 x^{2}-1}} = \sqrt{3}\sec^{-1}(3x) + C$$

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Most popular questions from this chapter

Shortcut for the Trapezoid Rule Prove that if you have \(M(n)\) and \(T(n)\) (a Midpoint Rule approximation and a Trapezoid Rule approximation with \(n\) subintervals), then \(T(2 n)=(T(n)+M(n)) / 2\).

Evaluate the following integrals. Consider completing the square. $$\int \frac{d x}{\sqrt{(x-1)(3-x)}}$$

Exact Simpson's Rule Prove that Simpson's Rule is exact (no error) when approximating the definite integral of a linear function and a quadratic function.

Use the indicated substitution to convert the given integral to an integral of a rational function. Evaluate the resulting integral. $$\int \frac{d x}{\sqrt[4]{x+2}+1} ; x+2=u^{4}$$

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