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Integrating 1/(sinx)^6 dx

Date: 02/25/2002 at 05:48:55
From: Ehab Nassif
Subject: Integration

What is the integration of 1/(sinx)^6 dx ?

Thanks a lot for your kindness.

Date: 02/26/2002 at 19:35:16
From: Doctor Jubal
Subject: Re: Integration

Hi Ehab,

Thanks for writing Dr. Math.

Here's a hint to get you started.

It would be a lot easier to integrate something like

  /  cosx dx
  | ---------
  / (sinx)^6

because then we could make the substitution u = sinx, du = cosx dx, 
and be left with an elementary integral. So it would be nice if we 
could somehow get a cosine in there.

To do this, note that

      1         (sinx)^2     1 - (cosx)^2
  ---------- = ---------- = --------------
   (sinx)^4     (sinx)^6       (sinx)^6


      1            1         (cosx)^2
  ---------- = ---------- + ----------
   (sinx)^6     (sinx)^4     (sinx)^6

and the integral becomes

  /    dx       /    dx       / (cosx)^2 dx
  | --------- = | --------- + | ------------
  / (sinx)^6    / (sinx)^4    /   (sinx)^6

Of these two integrals, the right one can be integrated by parts. How 
do you suppose the left one might be simplifed?

Does this help?  Write back if you'd like to talk about this some
more, or if you have any other questions.

- Doctor Jubal, The Math Forum   

Date: 03/03/2002 at 06:38:08
From: Ehab Nassif
Subject: Integration

Thanks a lot for your solution. I think I can proceed by myself with  
the rest of the steps.

Best regards,
Ehab Nassif
Associated Topics:
High School Calculus

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