1. ## Trigonometric integration

Hey, was wondering if anyone could help out with a pointer on how to do this in a more structured fashion.

So the original task is to solve the indefinite integral of
sqrt(1-x2)/x4

So taking a cue from the square root i substitute
x = sin(t)
dx = cos(t)dt

and get the integral of
cos2(t)/sin4(t)

And here my brain is going tilt, because I can't seem to figure out method will actually take me forward from there. What I did was working backwards, just playing around with trigonometric identities and finding that the derivative of
-1/3tan3(t)

gives the integrand. Back substituting, using that
tan(t) = sqrt(1-x2)

confirms that this is the correct answer.

But yea, this isn't a very elegant way of doing it, so maybe someone could point out what I'm missing?

2. ## Re: Trigonometric integration

It would help if you wrote out

$\dfrac{\cos ^2 (t)}{\sin^4 (t)} = \cot^2 (t) \csc^2 (t)$

So your integral will now be:

$\int \cot^2 (t) \csc^2 (t) dt$

If you let $u = \cot (t)$

you will find that $du = -\csc^2 (t) dt$

And then go from there...

3. ## Re: Trigonometric integration

Aaaaah. The sad thing is that I think I started out by trying that but got the secondary trig identities mixed up. Anyhows, many thanks for helping out!