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Math Help - Evaluate the Integral

  1. #1
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    Wink Evaluate the Integral

    Here is another integral Iím having trouble with...

    Evaluate:
    Attached Thumbnails Attached Thumbnails Evaluate the Integral-tal-1.jpg  
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  2. #2
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    Okay, let's do some of algebra:

    \begin{aligned}<br />
   \frac{x-5}{(x-4)(x-5)(x+5)}&=\frac{x+5-10}{(x-4)(x-5)(x+5)} \\ <br />
 & =\frac{1}{(x-4)(x-5)}-\frac{10}{(x-4)(x-5)(x+5)} \\ <br />
 & =\frac{1}{(x-4)(x-5)}-\underbrace{\frac{5}{4}\cdot \frac{(x-5)+(x+5)-2(x-4)}{(x-4)(x-5)(x+5)}}_{\alpha},<br />
\end{aligned}

    thus \alpha equals -\frac{5}{4}\left\{ \frac{1}{(x-4)(x+5)}+\frac{1}{(x-4)(x-5)}-\frac{2}{(x-5)(x+5)} \right\}, and the integrand equals

    \frac{1}{(x-4)(x-5)}-\frac{5}{4}\cdot \frac{1}{(x-4)(x+5)}-\frac{5}{4}\left\{ \frac{1}{(x-4)(x-5)}-\frac{2}{(x-5)(x+5)} \right\}, which can be rewritten as -\frac{1}{4}\frac{1}{(x-4)(x-5)}-\frac{5}{4}\left\{ \frac{1}{(x-4)(x+5)}-\frac{2}{(x-5)(x+5)} \right\}.

    Now, I'll you show how to ged rid of the remaining partial fraction descomposition, so we have \frac{1}{(x-4)(x-5)}=\frac{(x-4)-(x-5)}{(x-4)(x-5)}=\frac{1}{x-5}-\frac{1}{x-4} and you can do the same for the others. The rest is up to you and we're done!
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  3. #3
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Duffman View Post
    Here is another integral Iím having trouble with...

    Evaluate:
    Note that \int \frac {x - 5}{(x - 4)(x - 5)(x + 5)}~dx = \int \frac 1{(x - 4)(x + 5)}~dx = \frac 19 \int \left( \frac 1{x - 4} - \frac 1{x + 5} \right)~dx (via partial fractions or algebraic manipulaiton)
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  4. #4
    Math Engineering Student
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    Ahahaha, I just now realized that, haha, too brute force for me!!
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