Solving an Integral: Tips and Tricks for Evaluating ∫dx/(x√(x² - a²))

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Homework Help Overview

The discussion revolves around evaluating the integral ∫dx/(x√(x² - a²)), which falls under the subject area of calculus, specifically integration techniques involving rational functions and square roots.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • Participants discuss potential manipulations of the integral to relate it to known forms of integrals involving arccsch and arcsech. There is an attempt to suggest a substitution method, and one participant expresses realization about the integral's standard form.

Discussion Status

The discussion is active, with participants offering various approaches and hints. Some guidance has been provided regarding substitutions and recognizing the integral's form, but no consensus or complete solution has been reached.

Contextual Notes

Participants are navigating the challenge of transforming the integral into a recognizable format, with some expressing uncertainty about their previous knowledge and techniques.

insynC
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Homework Statement



Trying to evaluate the following integral:

∫dx/(x√(x² - a²))

The Attempt at a Solution



I think I'm missing something simple. I know:

∫dx/(x√(x² + a²)) = - 1/a arccsch|u/a| + C

&

∫dx/(x√(a² - x²)) = - 1/a arcsech(u/a) + C

But I'm not exactly sure how to manipulate my integral into one of these forms.

Any suggestions? Thanks
 
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\frac{1}{x(x^2-a^2)^{\frac{1}{2}}}

Does that help?
 
Let x = au, then dx = adu. Can you take it from there?
 
It's just a standard integral for arcsec... :S don't know how I missed that.

Thanks for the help
 

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