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Homework Help: Integration help please

  1. Jul 30, 2007 #1
    Integrating [tex]\frac{1}{(1+sinx)}[/tex]

    i just started learning Integration last week so not exactly sure how to approch this type of question.
     
  2. jcsd
  3. Jul 30, 2007 #2
    Do you know how to integrate 1/sinx?
     
  4. Jul 30, 2007 #3
    no this is the first time i have seen where a trigonometric function is on the denominator

    but now that i think about it

    sinx = [tex]\sqrt{1-cos^2x}[/tex]
    and arcsin was equal to [tex]\frac{1}{((1-x^2)}[/tex]

    so i guess i could use the U substituition method
    and then the answer would be...arcsin(cosx)??? im not too sure
     
    Last edited: Jul 30, 2007
  5. Jul 30, 2007 #4

    Dick

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    No, no arcsin. But you might want to try multiplying numerator and denominator by (1-sin(x)). It may look more familiar.
     
    Last edited: Jul 30, 2007
  6. Jul 30, 2007 #5

    daniel_i_l

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    You could also try the subsitution u = tan(x/2). It looks a little messy but everything cancels out.
     
  7. Jul 30, 2007 #6
    well im not sure if im doing the right thing but here goes....

    sinx = [tex]\frac{2tan(\frac{x}{2})}{1+tan^{2}(\frac{x}{2})}[/tex]

    soo then i symplify the equation so that it is

    [tex]\frac{1+tan^{2}(\frac{x}{2})}{1+tan^{2}(\frac{x}{2})+2tan(\frac{x}{2})}[/tex]

    the i used u = tan[tex](\frac{x}{2})[/tex]

    so i get [tex]\frac{1+u^{2}}{(u+1)^{2}}[/tex]

    What should i do from here or is this the right way at all?
     
  8. Jul 30, 2007 #7

    Dick

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    It looks to me like you are taking the long way around. Try multiplying numerator and denominator of your original problem by (1-sin(x)). You get (1-sinx)/cos^2(x). If you split that into two integrals you shouldn't have any problem with either of them.
     
  9. Jul 30, 2007 #8

    daniel_i_l

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    Dick solution is quicker in this case but to integrate things like
    1/(1+cosx+sinx) the substitution u=tan(x/2) is good.
    But notice that it isn't x = tan(u/2) but rather u=tan(x/2). In order to get this into something the the example I gave you can show with so trig identities that if u=tan(x/2) then:
    dx = 2du/(1+u^2)
    sinx = 2u/(1+u^2)
    cosx = (1-u^2)/(1+u^2)
    If you substitue all that you the (1+u^2)s candel out and you get some rational function which you can solve with the typical rational function method (breaking it into elementry functions...)
     
  10. Jul 30, 2007 #9
    wow by the way i got the answer using what dick said its actually pretty easy once u break it up...now im gonna try Daniels question gonna see if i can get those now :) thnx alot for the help by the way
    the answe was tanx - secx +c
     
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