20081001, 15:57  #1 
"Kyle"
Feb 2005
Somewhere near M52..
395_{16} Posts 
Challenging Integral
I was doing a homework problem involving the indefinate integral of sin(2x)/sqr(4cos(x) 1) but unable to find a way to solve it. I have tried several approaches involving a number of trig identities, but a solution still remains elusive. Can anyone give me a hint that I am probably missing? Thanks.

20081001, 17:19  #2  
Aug 2004
7×19 Posts 
Quote:
d/dx(cos(x)sqr(4cos(x)  1)) and follow your nose. Chris 

20081001, 19:33  #3 
Aug 2002
Ann Arbor, MI
1B1_{16} Posts 
Split the steps up line by line. You should only highlight the first parts first to see if you can complete it without seeing the whole solution.
Rewrite the numerator using sin(2x)=2*sin(x)*cos(x) Now do change of variables: w=4*cos(x)1 dw=4*sin(x) dx You should get 1/2*int[cos(x)/sqrt(w) dw] But you still have a term involving x, so you rearrange the substitution to get cos(x)=(w1)/4. This should give you 1/2*int[(sqrt(w)/4)(1/(4*sqrt(w))) dw], which is manageable. 
20081002, 01:05  #4 
"Kyle"
Feb 2005
Somewhere near M52..
7·131 Posts 
Thanks, I managed to solve it actually. Two substitutions, let u=4cos x, du = 4sin x dx. Then, let v=u 1, dv =du. It comes out very nicely.

20081002, 01:56  #5 
Aug 2002
Ann Arbor, MI
1B1_{16} Posts 

20081002, 06:03  #6  
Aug 2004
7×19 Posts 
Quote:
Code:
d/dx [ cos(x) sqrt(4cos(x)  1) ] = 2sin(x)cos(x) / sqrt(4cos(x)  1)
 sin(x) sqrt(4cos(x)  1)
=  sin(2x) / sqrt(4cos(x)  1)
 sin(x) sqrt(4cos(x)  1)
Code:
int[ sin(2x) / sqrt(4cos(x)  1) ] = cos(x) sqrt(4cos(x)  1)
 int [ sin(x) sqrt(4cos(x)  1) ]
= cos(x) sqrt(4cos(x)  1)
+ (4cos(x)  1)^(3/2) / 6 + constant


20081002, 09:28  #7 
Aug 2002
Ann Arbor, MI
433 Posts 
But integration by parts is a lot more work than substitution, especially when it's not at all obvious what your parts should be, and even more so when the result of integration by parts gives you another integral that a student not intimately familiar with the subject would still need to use substitution to evaluate ( int [ sin(x) sqrt(4cos(x)  1) ]).

20081221, 16:47  #8 
2^{3}×3×19^{2} Posts 
Integral Problem
The integration problem Sin(2x)/ Sqrt [4Cos(x) 1] has a solution according to Mathematica program. Since this is home work I cannot give you the answer

20081222, 15:37  #9 
Just call me Henry
"David"
Sep 2007
Liverpool (GMT/BST)
7·859 Posts 

20081223, 16:40  #10 
"Kyle"
Feb 2005
Somewhere near M52..
1110010101_{2} Posts 
If you absolutely want to, the answer is irrelevant at this point but would still be interesting.
Last fiddled with by Primeinator on 20081223 at 16:42 
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