Integrate (x^2)(sqrt(x^2-a^2))^(-1)

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SUMMARY

The integral of the function (x^2)(sqrt(x^2-a^2))^(-1) can be solved using trigonometric substitution. Specifically, substituting x = a sec(θ) simplifies the integral to one involving secant cubed. This method is well-documented, and resources such as the Wikipedia page on the integral of secant cubed provide comprehensive guidance. After performing the integration and back-substituting, the solution can be expressed as y = sqrt(x^2 - a^2).

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Anyone know how to integrate this?

Thanks
 
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Yes. Start by factoring out 1/a, then make the trig substitution x = a \sec \theta. After simplification, you will obtain an integral in terms of secant cubed, for which there is an entire http://en.wikipedia.org/wiki/Integral_of_secant_cubed" devoted to explaining how to integrate that. Do so, back-substitute, and simplify. Or alternatively, simply look this up in your handy table of integrals, and find that the solution is:

<< complete solution deleted by berkeman >>
 
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y=\sqrt{x^2-a^2} simplifies it a bit.
 

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