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Math Help - Partial fractions for integration

  1. #1
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    Partial fractions for integration

    Hi
    Can you tell me if my solution is correct?
    thanks
    John

    Evaluate using the method of partial fractions:

     \frac {10.dx}{(2x-6)(x+5)}

    This gives

     10=A.(x+5)+B.(2x-6)

    Making
     x=-5

     \frac{1}{-6}=B

     x=3

     \frac{5}{4}=A

    This results in an separate integrations of:

     5\int \frac{1}{4.(2x-6)}- \int \frac{1}{6(x+5)}

    This solves to give;

     \frac{5}{4} ln(2x-6)-\frac{1}{6}ln(x+5)+C
    Last edited by celtic1234; September 10th 2011 at 03:32 AM.
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  2. #2
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    Re: Partial fractions for integration

    What you are doing is correct but your write it in such a careless way that you give the wrong answer!
    Quote Originally Posted by celtic1234 View Post
    Hi
    Can you tell me if my solution is correct?
    thanks
    John

    Evaluate using the method of partial fractions:

     \frac {dx}{(x^2-1)}

    Below the line factors are

     \(x-1), (x+1)

     \int \frac {1}{(x^2-1)}.dx=\frac{A}{(x-1)}+\frac{B}{(x+1)}
    The integral sign should not be here. You mean simply
    \frac{1}{x^2- 1}= \frac{A}{x- 1}+ \frac{B}{x+1}

    This gives

     1=A.(x+1)+B.(x-1)

    Making
     x=-1

     \frac{1}{-2}=B

     x=1

     \frac{1}{2}=A

    This results in an separate integrations of:

     \int \frac{1}{2(x-1)}- \int \frac{1}{2(x+1)}

    This solves to give;

     \frac{1}{2} ln(x-1)-ln(x+1)+C
    No. The way you have written it, the "1/2" only multiplies the first ln and you want it with both:
    \frac{1}{2} ln(x-1)- \frac{1}{2}ln(x+1)+ C
    which is, of course, also equal to
    \frac{1}{2}ln(\frac{x-1}{x+1})+ C= ln\left(\sqrt{\frac{x-1}{x+1}}\right)+ C
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  3. #3
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    Re: Partial fractions for integration

    sorry i completely editing the latex and the problem?i have been studying for last 7 hours- i need to take a break....tired-somehow started to post a problem that i had the solution for? apologies
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  4. #4
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    Re: Partial fractions for integration

    Quote Originally Posted by celtic1234 View Post
    Hi
    Can you tell me if my solution is correct?
    thanks
    John

    Evaluate using the method of partial fractions:

     \frac {10.dx}{(2x-6)(x+5)}

    This gives

     10=A.(x+5)+B.(2x-6)

    Making
     x=-5

     \boxed{\frac{1}{-6}=B}<---------Incorrect. The correct answer is, B=-5/8

     x=3

     \frac{5}{4}=A

    This results in an separate integrations of:

     5\int \frac{1}{4.(2x-6)}- \int \frac{1}{6(x+5)}

    This solves to give;

     \frac{5}{4} ln(2x-6)-\frac{1}{6}ln(x+5)+C
    Please do not alter the original post after a solution is posted by someone. It causes confusion, specially for somebody seeing the thread after several posts are made. Always create new threads for new questions.
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  5. #5
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    Re: Partial fractions for integration

    thanks I see my mistake
    My problem is i am trying to post using Latex and need to edit the post after it's posted-by the time i edit it correctly somebody has replied....

    I presume the correct answer is;

     \frac{5}{4}ln(2x-6)-\frac{5}{8}ln(x+5)
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  6. #6
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    Re: Partial fractions for integration

    Quote Originally Posted by celtic1234 View Post
    thanks I see my mistake
    My problem is i am trying to post using Latex and need to edit the post after it's posted-by the time i edit it correctly somebody has replied....

    I presume the correct answer is;

     \frac{5}{4}ln(2x-6)-\frac{5}{8}ln(x+5)
    In obtaining the answer you have taken (which I did not notice in my previous post), \int\frac{dx}{2x-6}=\ln(2x-6) which is incorrect.

    I hope you know that the integration of the function \frac{1}{x+a} gives,

    \int\frac{dx}{x+a}=\ln|x+a|+C

    Hence, \int\frac{dx}{2x-6}=\frac{1}{2}\int\frac{dx}{x-3}=\frac{1}{2}\ln|x+3|

    Also you haven't put the modulus sign when taking the integrations.

    \frac{5}{4}\int \frac{dx}{2x-6}- \frac{5}{8}\int \frac{dx}{x+5}

    \frac{5}{8}\int \frac{dx}{x-3}- \frac{5}{8}\int \frac{dx}{x+5}

     \frac{5}{8}\left(\ln|x-3|-\ln|x+5|\right)+C

    \frac{5}{8}\ln\left|\frac{x-3}{x+5}\right|+C
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  7. #7
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    Re: Partial fractions for integration

    thanks Sudharaka-see my mistakes-too tired yesterday.....
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