How do I decompose this into partial fractions?

[stray crag]

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[shy oriole]

This isn't a rational function.
Rather, note that tan(x) = sin(x)/cos(x), so the whole thing is cos(x)/(a sin(x)^2).

So, if you want to integrate that, a substitution would work.

[tawdry wolf]

Alright thank you, this was something I got halfway through a u-substitution question, so I should do another substitution?

[fluid wolf]

Yes. Let u=sin(x).

[shy oriole]

What's the original question, by the way?

[pseudo flint]

Very neat sub

[shy oriole]

Ah, I see. You can also notice the following (for x > 0):
dx/(x^2 √(x^2 + a^2)) = dx/(x^3 √(1 + a^2/x^2)) = (-1/2)d(1/x^2)/√(1 + a^2/x^2)
And this is now an elementary integral.

[tawdry wolf]

Sorry, I'm kind of confused with the notation formatting on here. How would thatlast bit be written out?

[pseudo flint]

$-\int\frac{\frac12 d(\frac{1}{x^2})}{\sqrt{1+\frac{a^2}{x^2}}}$

[bronze basin]

mathisfun

[shy oriole]

Notice that the outlined differential is proportional to the differential of the expression under the root.

[tawdry wolf]

Thank you!

[shy oriole]

You're welcome!

[woeful fjord]

@tawdry wolf

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