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Math Help - stuck on this long division problem

  1. #1
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    stuck on this long division problem

    hi, this is as far as I got: stuck on this long division problem-long-division.jpg

    how do I integrate the rest? Also is my long division right?
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Your long division is o.k.

    Hint:

    Obtain the derivative of ln(x^2+4)...
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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     \int \frac{(4+x^3)}{(4+x^2)} dx<br />
    For the integrand \frac{(x^3+4)}{(x^2+4)}, do long division:

    =  \int  (x-(4 \frac{(x-1))}{(x^2+4)}) dx

    Integrate the sum term by term and factor out constants:

     =  \int x dx-4 \int \frac{(x-1)}{(x^2+4)} dx<br />
    Expanding the integrand \frac{(x-1)}{(x^2+4)} gives

    \frac{x}{(x^2+4)}-\frac{1}{(x^2+4)}:

     =  \int x dx-4 \int \frac{(x}{(x^2+4)}-\frac{1}{(x^2+4)} dx<br />
    Integrate the sum term by term and factor out constants:

     = 4 \int \frac{1}{(x^2+4)} dx-4 \int \frac{x}{(x^2+4)} dx+ \int x dx<br />
    For the integrand \frac{x}{(x^2+4)}}, substitute u = x^2+4 and du = 2 x dx:

     = -2 \int \frac{1}{u} du+4 \int \frac{1}{(x^2+4)} dx+ \int x dx<br />
    The integral of \frac{1}{(x^2+4)} is \frac{1}{2} arctan(\frac{x}{2}):

     = -2 \int \frac{1}{u} du+2 arctan(\frac{x}{2})+ \int x dx<br />
    The integral of \frac{1}{u} is ln(u):

     = -2 ln(u)+2 arctan(\frac{x}{2})+ \int x dx<br />
    The integral of x is \frac{x^2}{2}:

     = -2 ln(u)+\frac{x^2}{2}+2arctan(\frac{x}{2})+C<br />
    Substitute back for u = x^2+4:

     =\frac {x^2}{2}-2 ln(x^2+4)+2 arctan(\frac{x}{2})+C
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  4. #4
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    hi, i was doing good until

    the integral of is :

    how do you know this? If i did this problem, i would have ended up substituting u for (x^2+4)
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  5. #5
    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by softballchick View Post
    hi, i was doing good until

    the integral of is :

    how do you know this? If i did this problem, i would have ended up substituting u for (x^2+4)
    It's very famous integral...

    You can sole it by a trigonometric substitution: x=tan(u)
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