/*! 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 18 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 18

Evaluate the following integrals. $$\int \frac{d x}{\sqrt{x^{2}-49}}, x>7$$

Short Answer

Expert verified
Based on the provided step-by-step solution, the answer to the integral $$\int \frac{d x}{\sqrt{x^{2}-49}}, x > 7$$ is given by $$\ln|\frac{x}{7} + \frac{1}{7}\sqrt{x^2 - 49}| + C$$ where C is the integration constant.

Step by step solution

01

For the given integral, we can use the substitution formula: $$x = 7 \sec{\theta}$$ Now, we will find the differential \(dx\): $$dx = \frac{d(7\sec{\theta})}{d\theta} d\theta = 7 \sec{\theta} \tan{\theta} d\theta$$ #Step 2: Substitute x and dx into the integral#

Now substitute \(x\) and \(dx\) in the integral: $$\int \frac{dx}{\sqrt{x^2 - 49}} = \int \frac{7\sec{\theta}\tan{\theta} d\theta}{\sqrt{(7\sec{\theta})^2 - 49}}$$ #Step 3: Simplify the integral#
02

Simplify the integral using the trigonometric identity \(\sec^2{\theta} - 1 = \tan^2{\theta}\): $$\int \frac{7\sec{\theta}\tan{\theta} d\theta}{\sqrt{49(\sec^2{\theta} - 1)}} = \int \frac{7\sec{\theta}\tan{\theta} d\theta}{7 \sqrt{\tan^2{\theta}}} = \int \frac{\sec{\theta}\tan{\theta} d\theta}{\tan{\theta}}$$ The \(\tan{\theta}\) in the numerator cancels with the one in the denominator: $$\int \sec{\theta} d\theta$$ #Step 4: Integrate the function using Trigonometric Integration#

The integral of the secant function is calculated as follows: $$\int \sec{\theta} d\theta = \ln|\sec{\theta} + \tan{\theta}| + C$$ where \(C\) is the integration constant. #Step 5: Substitute back x and simplify the expression#
03

Now, we need to express our result in terms of \(x\). To do that, we will use the substitution we made in the beginning: $$x = 7\sec{\theta} \Rightarrow \sec{\theta} = \frac{x}{7}$$ We also need to find \(\tan{\theta}\) in terms of \(x\). We will use the trigonometric identity: $$\sec^2{\theta} - 1 = \tan^2{\theta} \Rightarrow \tan{\theta} = \sqrt{\sec^2{\theta} - 1} = \sqrt{\frac{x^2}{49} - 1} =\frac{1}{7}\sqrt{x^2 - 49}$$ Now, substitute \(\sec{\theta}\) and \(\tan{\theta}\) back into the antiderivative formula: $$\ln|\sec{\theta} + \tan{\theta}| + C = \ln|\frac{x}{7} + \frac{1}{7}\sqrt{x^2 - 49}| + C$$ #Step 6: Combine the constants and finish the solution#

Our final answer is: $$\int \frac{d x}{\sqrt{x^{2}-49}} = \ln|\frac{x}{7} + \frac{1}{7}\sqrt{x^2 - 49}| + C$$

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