Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question
Title: Evaluating Integrals Involving Square Roots --- **Problem Statement:** Evaluate the integral: \[ \int \frac{dx}{\sqrt{x^2 - 64}} \] where \( x > 8 \). --- **Solution Guide:** To solve this integral, note that it is of the form \(\int \frac{dx}{\sqrt{x^2 - a^2}}\), which can be evaluated using the inverse hyperbolic function formula. For the integral \(\int \frac{dx}{\sqrt{x^2 - a^2}}\), the solution is: \[ \ln |x + \sqrt{x^2 - a^2}| + C \] where \( C \) is the constant of integration. In this problem, \( a^2 = 64 \) which implies \( a = 8 \). Therefore, the integral becomes: \[ \int \frac{dx}{\sqrt{x^2 - 8^2}} \] Substituting \( a = 8 \) into the solution formula gives: \[ \ln |x + \sqrt{x^2 - 64}| + C \] Thus, the solution to the integral is: \[ \ln |x + \sqrt{x^2 - 64}| + C \] Remember to add the constant of integration \( C \) as this is an indefinite integral. --- **Conclusion:** This problem illustrates the method for evaluating integrals involving square root expressions of the form \(\sqrt{x^2 - a^2}\) using inverse hyperbolic functions. This is a common technique applicable in various fields of mathematics, including calculus and engineering.
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Transcribed Image Text:Title: Evaluating Integrals Involving Square Roots --- **Problem Statement:** Evaluate the integral: \[ \int \frac{dx}{\sqrt{x^2 - 64}} \] where \( x > 8 \). --- **Solution Guide:** To solve this integral, note that it is of the form \(\int \frac{dx}{\sqrt{x^2 - a^2}}\), which can be evaluated using the inverse hyperbolic function formula. For the integral \(\int \frac{dx}{\sqrt{x^2 - a^2}}\), the solution is: \[ \ln |x + \sqrt{x^2 - a^2}| + C \] where \( C \) is the constant of integration. In this problem, \( a^2 = 64 \) which implies \( a = 8 \). Therefore, the integral becomes: \[ \int \frac{dx}{\sqrt{x^2 - 8^2}} \] Substituting \( a = 8 \) into the solution formula gives: \[ \ln |x + \sqrt{x^2 - 64}| + C \] Thus, the solution to the integral is: \[ \ln |x + \sqrt{x^2 - 64}| + C \] Remember to add the constant of integration \( C \) as this is an indefinite integral. --- **Conclusion:** This problem illustrates the method for evaluating integrals involving square root expressions of the form \(\sqrt{x^2 - a^2}\) using inverse hyperbolic functions. This is a common technique applicable in various fields of mathematics, including calculus and engineering.
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Step 1: Given:

I equals integral fraction numerator d x over denominator square root of x squared minus 64 end root end fraction comma space x greater than 8

We have to evaluate the given integration.

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