Integral of csc^4(x)*cot^6x dx

I'm not sure how to substitute it in a way that makes sense. I tried making it 1/sin^4 * cos^6/sin^6 = cos^6 / sin^10 but I got stuck, and I'm not sure it's the right way to go.

Question

Integral of csc^4(x)*cot^6x dx

I'm not sure how to substitute it in a way that makes sense. I tried making it 1/sin^4 * cos^6/sin^6 = cos^6 / sin^10 but I got stuck, and I'm not sure it's the right way to go.

e.mccormick · Answer

hmmm....  $\csc^4=\csc^2\cdot \csc^2 = \csc^2(\cot^2+1)$

That get you anywhere?

theoreo · Answer

Wow, I think I was really over thinking this one. 

So since $$\csc ^2 = (\cot^2+1)$$ 

$$\int\limits \csc^4*\cot^6 = (\cot^2+1)^2*\cot^6$$

Then I can just make u= cot and integrate that. Right?

e.mccormick · Answer

Something like that.  Should get you there with a start like that.  There are a couple ways I think it could be done but it has been a while since I used my calculus.  Hehe.

theoreo · Answer

OK thanks a lot.

e.mccormick · Answer

np.  Have fun.