Results 1 to 5 of 5

Math Help - Need help with an integral please

  1. #1
    Member
    Joined
    Sep 2009
    Posts
    242
    Thanks
    1

    Need help with an integral please

    Please help me to integrate...

    \int \frac{N}{2}\sigma\sin{(\omega \ln{\sigma })}+\frac{N}{2}\sigma\; d\sigma

    This is how far I've gotten...

    \frac{N}{2}\int \sigma\sin{(\omega \ln{\sigma })}+\sigma\; d\sigma

    u\equiv \sin(\omega \ln{\sigma})

    du = \frac{\omega \cos(\omega \ln{\sigma})}{\sigma}\; d\sigma

    d\sigma = \frac{\sigma}{\omega \cos(\omega \ln{\sigma})}\; du

    \frac{N}{2}\int \frac{u\sigma^2}{\omega \cos(\omega \ln{\sigma})}\; du +\frac{N}{2}\int \sigma\; d\sigma

    ...not sure what to do...sos...sos...
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Jan 2009
    Posts
    715
    Quote Originally Posted by rainer View Post
    Please help me to integrate...

    \int \frac{N}{2}\sigma\sin{(\omega \ln{\sigma })}+\frac{N}{2}\; d\sigma

    This is how far I've gotten...

    \frac{N}{2}\int \sigma\sin{(\omega \ln{\sigma })}+1\; d\sigma

    u\equiv \sin(\omega \ln{\sigma})

    du = \frac{\omega \cos(\omega \ln{\sigma})}{\sigma}\; d\sigma

    d\sigma = \frac{\sigma}{\omega \cos(\omega \ln{\sigma})}\; du

    \frac{N}{2}\int \frac{u\sigma^2}{\omega \cos(\omega \ln{\sigma})}\; du +\frac{N}{2}\int 1\; d\sigma

    ...not sure what to do...sos...sos...

    There are three methods : the first one also the most effective one i suggest is sub.  \ln{\sigma} = t or  \sigma = e^t

     d\sigma = e^t~dt so we have

     \frac{N}{2} \int (e^t)\sin(\omega t) (e^t~dt)

     = \frac{N}{2} \int e^{2t} \sin(\omega t )~dt

    Definitely , it is an integral which is more familiar to you , you can solve it yourself now


    The second method is using integration by parts , you can discover how it can work .

    The third is the most powerful but it requires much attention :


     \int x\sin(\omega\ln{x})~dx

     = lm\left\{\ \int x e^{i \omega \ln{x} }~dx \right\}\

     = lm\left\{\ \int x e^{ \ln{x ^{i \omega }} }~dx \right\}\

     = lm\left\{\ \int x \cdot x^{i \omega }~dx \right\}\

     = lm\left\{\ \int x^{1+ i \omega }~dx \right\}\

     = lm[ \frac{x^{2+i\omega} } { 2+i\omega} ] + C

     = lm[ \frac{x^2 exp \{\ \ln{ x^{i\omega}} \}\ ( 2 - i\omega )}{ 4 + \omega^2 } ] + C

     = \frac{ x^2 ( 2\sin(\omega\ln{x}) - \omega \cos(\omega\ln{x}))}{ 4 + \omega^2 } + C
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,404
    Thanks
    1293
    Quote Originally Posted by rainer View Post
    Please help me to integrate...

    \int \frac{N}{2}\sigma\sin{(\omega \ln{\sigma })}+\frac{N}{2}\sigma\; d\sigma

    This is how far I've gotten...

    \frac{N}{2}\int \sigma\sin{(\omega \ln{\sigma })}+\sigma\; d\sigma

    u\equiv \sin(\omega \ln{\sigma})

    du = \frac{\omega \cos(\omega \ln{\sigma})}{\sigma}\; d\sigma

    d\sigma = \frac{\sigma}{\omega \cos(\omega \ln{\sigma})}\; du

    \frac{N}{2}\int \frac{u\sigma^2}{\omega \cos(\omega \ln{\sigma})}\; du +\frac{N}{2}\int \sigma\; d\sigma

    ...not sure what to do...sos...sos...
    http://www.wolframalpha.com/input/?i...5D+%2B+N%2F2*x

    Click Show Steps
    Last edited by mr fantastic; June 9th 2010 at 01:20 AM. Reason: Fixed link.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Oct 2009
    Posts
    35
    Quote Originally Posted by rainer View Post
    Please help me to integrate...

    \int \frac{N}{2}\sigma\sin{(\omega \ln{\sigma })}+\frac{N}{2}\; d\sigma

    This is how far I've gotten...

    \frac{N}{2}\int \sigma\sin{(\omega \ln{\sigma })}+1\; d\sigma
    u \equiv \ln{\sigma}

    du = 1/\sigma d\sigma implies d\sigma = \sigma du

     \frac{N}{2} \int e^{2u} \sin(\omega u)du + \frac{N}{2}\int 1\; d\sigma

     \frac{Ne^{2u}\sin(\omega u)}{4} + \frac{N}{2} \int e^{2u} \omega \cos(\omega u)du + \frac{N\sigma}{2}

    \frac{N\sigma ^2}{4}(\sin (\omega \ln{\sigma})+ \omega \cos (\omega \ln{\sigma}))- \frac{N\omega ^2}{2}\int e^{2u}\sin(\omega u)du ~+  \frac{N\sigma}{2} = \frac{N}{2} \int e^{2u} \sin(\omega u)du + \frac{N\sigma}{2}

    \frac{N}{2}(1+\omega ^2) \int e^{2u} \sin(\omega u)du = \frac{N\sigma ^2}{4}(\sin (\omega \ln{\sigma})+ \omega \cos (\omega \ln{\sigma}))

    \int \sigma\sin{(\omega \ln{\sigma })} d\sigma = \frac{\sigma ^2(\sin (\omega \ln{\sigma})+ \omega \cos (\omega \ln{\sigma}))}{2+2\omega ^2}

    \frac{N}{2}\int \sigma\sin{(\omega \ln{\sigma })}+1\; d\sigma = \frac{N \sigma ^2(\sin (\omega \ln{\sigma})+ \omega \cos (\omega \ln{\sigma}))}{4+4\omega ^2} + \frac{N\sigma}{2} + C

    ...which doesn't agree with mathematica. I should not be allowed to answer integral questions at 1 am :/
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Sep 2009
    Posts
    242
    Thanks
    1
    Quote Originally Posted by simplependulum View Post
     = \frac{N}{2} \int e^{2t} \sin(\omega t )~dt

    Definitely , it is an integral which is more familiar to you , you can solve it yourself now
    Oh yes, yes indeed.

    Quote Originally Posted by simplependulum View Post
    The third is the most powerful but it requires much attention
    Over my head for now, but very interesting. At first I did the integral by substituting euler's relation for the sine function. That made the integral very easy going, but the result was totally useless to me. "i" and me don't get along very well.

    Quote Originally Posted by Turiski View Post
    ...which doesn't agree with mathematica. I should not be allowed to answer integral questions at 1 am :/
    Turiski- The magic recipe for integrating \int e^{2t} \sin(\omega t )~dt has a minus sign where you have a plus sign. It also has the exponential factor 2 down in front of sin(wln(s)). Otherwise looks like you were on target.

    Thanks everybody!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: August 31st 2010, 07:38 AM
  2. Replies: 1
    Last Post: June 2nd 2010, 02:25 AM
  3. Replies: 0
    Last Post: May 9th 2010, 01:52 PM
  4. [SOLVED] Line integral, Cauchy's integral formula
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: September 16th 2009, 11:50 AM
  5. Replies: 0
    Last Post: September 10th 2008, 07:53 PM

Search Tags


/mathhelpforum @mathhelpforum