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!

Integral of 1/u^n knowing integral of 1/u

  1. Dec 9, 2012 #1
    Hello,

    This is the initial problem:

    Find the primitive of 1/(x4+1)

    i've done this, the value of the primitive is this ugly looking expression:

    http://image.bayimg.com/d9ac5052a83d5808955a7a647c6cb7343f9ace1f.jpg [Broken]

    Now the question asked is to deduce the primitive value of 1/(x4+1)^3 from what I found.

    This is why I'm asking if there is any general method to compute the value of ∫1/u^n knowing ∫1/u

    Thank you
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Dec 9, 2012 #2

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    No, essentially there isn't. [itex] \int \frac{1}{(1+x^4)^3} [/itex] is solved through partial fractions.
     
  4. Dec 10, 2012 #3

    Mark44

    Staff: Mentor

    Your image isn't showing.
    Two comments here:
    1. The two formulas are not related.
    $$ \int \frac{du}{u^n} = \int u^{-n}du = \frac{u^{-n + 1}}{-n + 1} + C$$
    $$\int \frac{du}{u} = ln|u| + C$$

    The integrals you show don't fit either of these formulas, because if u = x4 + 1, then du = 4x3dx, which you don't have.

    2. You have omitted the differential du from your integrals above. This isn't so crucial in the early stages of learning integration, but it is for more complicated methods such as integration by parts and trig substitution.
     
    Last edited by a moderator: May 6, 2017
  5. Dec 10, 2012 #4
    Thanks for the answers.

    I'm going to try rewriting 1/(x4+1) in partial equations and see what I can get.
     
  6. Dec 10, 2012 #5

    Mark44

    Staff: Mentor

    Presumably you mean 1/(x4 + 1). At the very least, use '^' to write this as 1/(x^4 + 1).

    As for breaking up 1/(x4 + 1) using partial fractions, I don't see how this will do you any good. The denominator cannot be reduced to lower-degree factors with real coefficients.

    It would help if you showed us the exact problem you're working on.
     
  7. Dec 10, 2012 #6

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    x4 + 1 can be reduced to lower-degree factors with real coefficients, but not to lower-degree factors with integer coefficients.

    [itex]x^4+1=(x^2+(\sqrt{2})x+1)(x^2-(\sqrt{2})x+1)[/itex]
     
  8. Dec 10, 2012 #7
    This problem is presented in 2 questions.

    1) Find the primitive of [itex]\frac{1}{x^4 + 1}[/itex]

    This is done, no particular problem. You get this:

    http://image.bayimg.com/d9ac5052a83d5808955a7a647c6cb7343f9ace1f.jpg [Broken]


    2) Deduce the primitive of [itex]\frac{1}{(x^4 + 1)^3}[/itex] from the primitive of [itex]\frac{1}{x^4 + 1}[/itex]


    Now the problem is with 2). I don't see how I can use the first result to find the second one.

    Thank you for your time.
     
    Last edited by a moderator: May 6, 2017
  9. Dec 10, 2012 #8

    Mark44

    Staff: Mentor

    Thanks...
     
  10. Dec 10, 2012 #9

    Mark44

    Staff: Mentor

    The image above is attached here.
    d9ac5052a83d5808955a7a647c6cb7343f9ace1f.jpg
     
    Last edited by a moderator: May 6, 2017
  11. Dec 10, 2012 #10

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    Last edited: Dec 10, 2012
  12. Dec 10, 2012 #11

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    Integration by parts once is enough to determine a recursive relationship of the form [itex]I_{n+1}(x)=f(x, n)I_n(x)+g(x, n)[/itex]. [itex]I_3[/itex] can then be expressed in terms of [itex]I_1(x)[/itex] by using that relationship twice.
     
  13. Dec 10, 2012 #12
    Thank you, this looks promising and within my reach.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Integral of 1/u^n knowing integral of 1/u
  1. Integral u^2/(1+u^4) (Replies: 5)

Loading...