1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    SteamKing

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

    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

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    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

    ln|√u^2+1|

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

    ln|√(x+1)^2+1|

    Final answer

    ln|√(x+1)^2+1| - arctan(x+1) + C

    Correct ?
     
  7. Jul 30, 2014 #6

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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

    BiGyElLoWhAt

    User Avatar
    Gold Member

    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

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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 ?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Need Help With This Integral
Loading...