May 16th, 2007, 04:43 AM  #1 
Newbie Joined: May 2007 From: Belgium Posts: 4 Thanks: 0  Integration problems
hi, I can't seem to find the solution to these exercises 1) INT (cos(x))^7 dx 2) INT (x/(1+x^6)) dx (I tried to do something with tan^1(x) but no success) thanks 
May 16th, 2007, 09:33 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,757 Thanks: 2138  Re: Integration problems
1) ?(cos x)^7 dx Hint: use integration by parts, then replace sin²x with 1  cos²x. 2) ?(x/(1+x^6)) dx Hint: 1 + x^6 ? (1 + x²)(1  x² + x^4), so use carefully chosen partial fractions. 
May 16th, 2007, 02:35 PM  #3 
Newbie Joined: May 2007 From: Belgium Posts: 4 Thanks: 0 
thx, I tried to use those hints, maybe i've come a litle further now. I still can't see it tough hope you can read my writings (im quite bad in having some order :P ) 1) now i only seem to have gotten the same int + more 2) I get left with the circled equation for which I don't find a method to solve it 
May 16th, 2007, 06:51 PM  #4 
Senior Member Joined: Dec 2006 Posts: 1,111 Thanks: 0 
Try trigonometric substitution on the second problem. Set up your triangle with a 1 on the bottom, an x^3 on the upright side, and a ?(1 + x^6) on the hypotenuse. The angle is in the lower left corner. x^3 = tan ? x = (tan ?)^(1/3) ?(1 + x^6) = sec ? 1 + x^6 = (sec ?)² x^3 = tan ? 3x² dx = (sec ?)² d? dx = (sec ?)² / 3x² d? dx = (sec ?)² / (3(tan ?)^(2/3)) d? Thus, ?x/(1+x^6) dx Is equivalent to: ?(tan ?)^(1/3) * (sec ?)²/ ((sec ?)² * (3(tan ?)^(2/3))) d? Cancel terms and pull out the constants: (1/3) * ?1/(tan ?)^(1/3) d? (1/3) * ?(cot ?)^(1/3) d? Well, that doesn't seem like quite so bad of a problem to integrate. See if you can integrate that, and then backsubstitute to transform it into an answer in terms of 'x'. 
May 17th, 2007, 01:54 AM  #5 
Newbie Joined: May 2007 From: Belgium Posts: 4 Thanks: 0 
hi, thanks for your help, unfortunately I already tried that road and I too get left with the equation you write there, it seems to me its not so easy to solve at all If i write the integral in an integrator ( http://integrals.wolfram.com/index.jsp ) I get the following solution which looks prety hard to find, maybe with integration by parts, I'll keep trying 
May 17th, 2007, 04:40 AM  #6 
Senior Member Joined: Dec 2006 Posts: 1,111 Thanks: 0 
Hmm... Have you tried playing with hyperbolic trig functions yet? It almost seems to me that they're somehow splitting up the original problem into a trig and a hyperbolic trig integral. 
May 17th, 2007, 11:21 AM  #7  
Global Moderator Joined: Dec 2006 Posts: 20,757 Thanks: 2138  Quote:
 
May 17th, 2007, 07:38 PM  #8  
Member Joined: Dec 2006 From: St. Paul MN USA Posts: 37 Thanks: 0  Re: Integration problems Quote:
 
May 18th, 2007, 03:52 AM  #9 
Newbie Joined: May 2007 From: Belgium Posts: 4 Thanks: 0 
thanks for all the help, I think I found the first one now , I'll try the second one this eve hope it's correct... 
May 22nd, 2007, 10:02 PM  #10 
Newbie Joined: May 2007 Posts: 7 Thanks: 0  There is another way to solve the problem 

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