1. Not finding help here? Sign up for a free 30min 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 x*sin(ax)

  1. Nov 29, 2004 #1
    What is the integral of x*sin(x) and x*sin(ax)?
    I have no idea since I have neveer integrated something to get a product...
    Ohh, it's supposed to be integrated from 0 to 1 for the sin(ax)
  2. jcsd
  3. Nov 29, 2004 #2


    User Avatar
    Staff Emeritus
    Science Advisor

    Use "integration by parts".

    From the product rule for derivatives, d(uv)/dx= u(dv/dx)+ v(du/dx). We can write that in "differential" form as d(uv)= u dv+ vdu and then rewrite it as

    u dv= d(uv)- vdu.

    Integrating both sides gives the integral formula
    [tex]\int u dv= uv- \int vdu[/tex].

    In particular, to integrate x sin(ax), let u= x, dv= sin(ax) dx. Then du= dx and
    v= -(1/a)cos(ax) so
    [tex]\int x sin(ax)dx= -(\frac{1}{a}x cos(ax)+ \frac{1}{a}\int cos(ax) dx[/tex]

    [tex]= -\frac{1}{a}(x cos(x)+ \frac{1}{a}sin(ax))[/tex].
  4. Nov 29, 2004 #3


    User Avatar
    Science Advisor
    Homework Helper

    Sorry,there's a minus,a paranthesis too much and an "a" missing:
    [tex]\int x sin(ax)dx= -\frac{1}{a}x cos(ax)+ \frac{1}{a}\int cos(ax) dx[/tex]

    [tex]= -\frac{1}{a}[x cos(ax)- \frac{1}{a}sin(ax)][/tex]
  5. Nov 29, 2004 #4
    and if its / ? =) as in sin(ax)/x
    Or is it so easy that I can do it by myself, don't have time right now...
  6. Nov 29, 2004 #5
    This is what I seem to get, very annoying
    [tex]\int(sin(ax)\frac{1}{x})=sin(ax) ln(x)+\int(ln(x) cos(ax))[/tex]

    Any ideas? Are any of the following integrals easy to do?
  7. Nov 29, 2004 #6
    did u try setting u=x and dv=sin(ax)dx ?
    this is what i got
    [tex] \int xsin(ax) dx=-\frac{x}{a}cos(ax) +\frac{1}{a}\int cos(ax) dx[/tex]
    [tex] = -\frac{x}{a}cos(ax)+\frac{1}{a^2}sin(ax)[/tex]
  8. Nov 30, 2004 #7
    heh vladimir, I understand the xsin(ax) integral but now I am trying to do 1/x*sin(ax) is this possible? If you look at my previous post you'll see me trying to integrate 1/x*sin(ax)
  9. Nov 30, 2004 #8


    User Avatar
    Science Advisor
    Homework Helper

    That's because there is no primitive of the function sinx/x.
    I assume you know that ordinary functions can be differentiated and the result be another "familiar" function.But this does not apply for primitives.There are functions like sinx/x,cos/x,exp(x^2),etc. which do not have primitives.That is,u cannot find a function which to differentiate to get the function you wish to integrate.
    However,numerical methods based on Taylor/Mac Laurin formula(s) can be used to obtain results.For example,to find the primitive of sinx/x,u need to expand sinx and devide each term of the expansion term by x and integrate the results.You'll have then a new infinite series,which could be seen as the Taylor/Mac Laurin exapansion of the function u are looking for.
    This thing works for functions which "behave" pretty well as to apply Taylor/Mac Laurin formula(s) to them.The 3 examples i have stated prove this assertion.
    To find definite integral values for the 3 functions mentioned above,try to get a hand on 2 books:M.Abramowitz,I.Segun:"Mathematical functions and tables" and Rytzhik and Gradstein:"Tables of integrals" and search for sine integral function,cosine integral function and erf(error) function.

    P.S.I'm not at the library anymore,so from now on,when i give indications to certain books always doubt the veridicity of the names and titles stated,as i give them from my memory to which i have no recollection of having ever been treated with glucosis.So it cold fail me someday.Hopefully not soon.
    Last edited: Nov 30, 2004
  10. Dec 1, 2004 #9
    Thanks man, I was expecting the likes...
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?