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November 27th, 2008, 05:55 PM  #1 
Newbie Joined: Nov 2008 Posts: 7 Thanks: 0  Determining bounds when changing variables
Hello everybody! I have this question. Say I want to integrate sqrt(x^21) between 0 and sqrt(3)/2. I can substitute x=cos(x) to integrate sin^2(x) between cos^(1)(0) and cos^(1)(sqrt(3)/2). But how do I choose which values to use? The integral of sin^2(x) is (x  sinx cosx)/2. Obviously taking cos^(1)(0) = 3pi/2 or cos^(1)(0) = pi/2 will give different answers. Now obviously cos^(1)(x) is typically understood to give an answer in the range [0, pi]. But why wouldn't the integration work with the bound 3pi/2? I am a bit puzzled. Thanks a lot! Isaac P.S. Why not implement LaTeX with this forum? 
November 27th, 2008, 06:12 PM  #2  
Senior Member Joined: Jul 2008 Posts: 895 Thanks: 0  Re: Determining bounds when changing variables Quote:
2. To be more direct, if x = cos A, then A = cos^(1)A. So, if x = 0, A = ? If x = sqrt(3)/2, A = ? You need to be familiar already with basic trig functions to do those. 3. Latex does work here. You should start over by drawing a right triangle with angle A, x on the hypotenuse, and 1 on the adjacent sides. Then sqrt(x^21) will be the other side, then take it from there.  
November 27th, 2008, 06:26 PM  #3 
Newbie Joined: Nov 2008 Posts: 7 Thanks: 0  Re: Determining bounds when changing variables
Thank you for your answer. Obviously I should have written . So we have Now obviously the answer will depend on the value chosen for the bounds. My question is, must the bounds be chosen in the range , and, if so, why? 
November 28th, 2008, 08:18 AM  #4 
Newbie Joined: Nov 2008 Posts: 16 Thanks: 0  Re: Determining bounds when changing variables
if you want to evaluate the integral in terms of u then you need to change the bounds. if you plan on substituting the u's back into x's then you can use the original bounds since the integral was really from x=0 to x=sqrt(3)/2.

November 28th, 2008, 08:48 AM  #5  
Newbie Joined: Nov 2008 Posts: 7 Thanks: 0  Re: Determining bounds when changing variables Quote:
 
November 28th, 2008, 11:28 AM  #6 
Newbie Joined: Nov 2008 Posts: 16 Thanks: 0  Re: Determining bounds when changing variables
I would try to do a substitution involving the identity tan^2(x) = sec^2(x)  1

November 28th, 2008, 12:59 PM  #7 
Senior Member Joined: Jul 2008 Posts: 895 Thanks: 0  Re: Determining bounds when changing variables
Just a note on the bounds, taking you back to those earlier studies as I suggested: Draw a square and cut it in half with a diagonal. You then have a 45,45,90 triangle with sides in the ratio 1:1:sqrt(2). Draw an equilateral triangle, and cut it in half with an altitude. You then have a 30,30,90 triangle with sides in the ratio 1:2:sqrt(3). Label the angles and find the trig ratios of sides to answer the question about what the angles are which are the bounds we are talking about here. 

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