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Homework Help: Need Help With This Integral

  1. Jul 30, 2014 #1
    1. The problem statement, all variables and given/known data

    Hi guys, I need help on how to go about solving this integral

    ∫x/(x^2+2x+2) dx

    2. Relevant equations

    3. The attempt at a solution

    My math professor said something about "force the derivative of denominator to occur at top and compensate" but I have no idea what he means. Can somebody please help me, thanks
  2. jcsd
  3. Jul 30, 2014 #2


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    What's the derivative of the denominator? Is it similar to the numerator?
  4. Jul 30, 2014 #3
    It's 2x+2, do you mean I should use u substitution ?
  5. Jul 30, 2014 #4


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    I would start by completing the square in the denominator- [itex]x^2+ 2x+ 2= x^2+ 2x+ 1+ 1= (x+1)^2+ 1[/itex]
    [tex]\int \frac{x}{x^2+ 2x+ 2} dx= \int \frac{x}{(x+ 1)^2+ 1} dx[/tex]

    Now let u= x+ 1 so that du= dx and x= u- 1.
  6. Jul 30, 2014 #5
    Thanks, this is what I have tried, correct me if I'm wrong

    by completing the square and u sub

    ∫(u-1)/(u^2+1) du

    = ∫ u/(u^2+1) - ∫1/(u^2+1)

    For the first integral I used a trig substitution, like this

    u = tan(t)
    du = sec(t)^2 dt

    plugging back in I get,

    ∫(tan(t)*sec(t)^2)/(sec^2(t)) dt...................tan(t)^2 + 1 = sec(t)^2

    = ∫tan(t) dt
    = ln |sec(t)|

    Using the triangle thing, I get


    and, u = x+1, so plugging back in


    Final answer

    ln|√(x+1)^2+1| - arctan(x+1) + C

    Correct ?
  7. Jul 30, 2014 #6


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    Here's what your prof was suggesting:$$
    \int\frac x {x^2+2x+2}~dx = \frac 1 2 \int\frac {2x}{x^2+2x+2}~dx
    =\frac 1 2 \int\frac{(2x+2)-2}{x^2+2x+2}~dx =\frac 1 2\int\frac{2x+2}{x^2+2x+2}~dx
    -\int \frac 1 {x^2+2x+2}$$Now the first integral is set up for the natural u substitution and the second is ready to complete the square as others have suggested. Personally, I would just complete the square in the first place as Halls has suggested. Your prof's method is sometimes useful.
  8. Jul 30, 2014 #7
    Thanks, I'm a bit confused though

    Where did the 1/2 come from ?

    and, the derivative of x^2+2x+2 = 2x+2, so where did the 2x come from ?
  9. Jul 30, 2014 #8


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    the 1/2 is a "multiply by 1" thing. Multiply x by 2 and then the whole fraction (effectively the x again) by 1/2.
  10. Jul 30, 2014 #9


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    You are trying to build a ##2x+2## in the numerator so you start by writing ##x = \frac 1 2\cdot 2x##. Then you add and subtract ##2##. That's what your prof meant by forcing the derivative of the denominator into the numerator.
  11. Jul 30, 2014 #10
    Oh, yes I see, but isn't that suppose to be 2x+2 (referring to integral with the 2x) as that's the derivative of the denominator ?
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