Is success realistic using u-substitution when the derivative isn't in the integrand?

Metronome

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Jun 12, 2018
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If one attempts a u-substitution that leaves x in the new integrand, it stands to reason that the x could be eliminated by inverting the substitution function and making another substitution. For example, to take the integral of sqrt(1 + x^-2) dx, we might let u = 1 + x^-2, and thus dx = -((x^3)/2) du. The u-substitution yields integral of -sqrt(u)((x^3)/2) du. We could solve for u in terms of x by inverting the substitution function to get x =±sqrt(1/(u - 1)), and eliminate the final reference to x the integrand. However, I have never had any luck getting a nicer integral at the end of this process. Is there some reason this approach is doomed to fail (due to the linear dependence of the two substitutions or something), or is always worth trying?
 
If one attempts a u-substitution that leaves x in the new integrand, it stands to reason that the x could be eliminated by inverting the substitution function and making another substitution. For example, to take the integral of sqrt(1 + x^-2) dx, we might let u = 1 + x^-2, and thus dx = -((x^3)/2) du. The u-substitution yields integral of -sqrt(u)((x^3)/2) du. We could solve for u in terms of x by inverting the substitution function to get x =±sqrt(1/(u - 1)), and eliminate the final reference to x the integrand. However, I have never had any luck getting a nicer integral at the end of this process. Is there some reason this approach is doomed to fail (due to the linear dependence of the two substitutions or something), or is always worth trying?
Try trigonometric substitution of

1/x = tan(Θ)

This substitution will make it a "little" easier.
 
Thanks, I just used the example as an illustration though. I'm wondering how often if ever u-substitution works out by inverting the substitution function to eliminate additional references to x added to the integrand when replacing dx with du.
 
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