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Some very tough integration problems I am stuck on.

  1. Feb 7, 2008 #1
    [Solved] Some very tough integration problems I am stuck on.

    Got these for homework the other day and they are due today! I am having trouble on where to start with all of them.

    [tex]\int x^{n}lnx dx[/tex]

    [tex]\int sin(\sqrt{x}) dx[/tex]

    don't even know where to start on this one... should I substitute for sqrt(x)?

    [tex]\int xarctan(x^{2}-1) dx[/tex]

    I think I should substitute for what's inside arctan, but then i'm left with [tex]\frac{1}{2} \int arctan(u) dx[/tex] and I don't know how to integrate arctan
     
    Last edited: Feb 7, 2008
  2. jcsd
  3. Feb 7, 2008 #2

    Hootenanny

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    I'll give you a hint for the first one: Integration by parts, but be careful which one you chose to differentiate (which one simplifies on differentiation?).
     
  4. Feb 7, 2008 #3
    Ok that's what I thought for the first one. We were taught to pick the one that wont change for the part that we integrate... and the other for the part that we differentiate.

    So I would pick lnx to differentiate? OHH!!! I just figured the first one out! Thanks so much!

    Still stuck on the others though.
     
  5. Feb 7, 2008 #4

    Hootenanny

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    Well done :approve:

    For the second one, how about a substitution (followed by integration by parts)?
     
  6. Feb 7, 2008 #5
    Ok This is what I did for the second one:

    I picked sqrt(x) as u.

    [tex]2 \int usin(u) du[/tex]

    The integration by parts is what I has me confused. I know this is the form:

    [tex]uv - \int v du[/tex]

    I'm not sure what to pick as my u and dv, my teacher didn't explain it very well in class. I think I should pick sin(u) for my dv though and the rest for my u (or in this case t since I already used u for my substitution). Am I on the right track?

    Thanks for all of your help by the way. :)
     
  7. Feb 7, 2008 #6
    well, generally there is no one pattern on how to approach these problems, i mean on which one to chose as v and which as du, it comes with experience. Meanwhile some trial and error would be just fine.
     
  8. Feb 7, 2008 #7
  9. Feb 7, 2008 #8
    Wow, perfect thank you very much. Wish my teacher would have shown me that one.
     
  10. Feb 7, 2008 #9
    Here are my solutions for the first two:

    [tex]1. \left[ \frac{xlnx}{n+1} - \frac{x}{(n+1)^{2}} \right] x^{n}[/tex]

    [tex]2. 2sin(\sqrt{x}) - 2\sqrt{x}cos(\sqrt{x})[/tex]

    I think that's right for #2

    3. Still no go
     
  11. Feb 7, 2008 #10
    Here are my solutions for the first two:

    1. [tex]\left[ \frac{xlnx}{n+1} - \frac{x}{(n+1)^{2}} \right] x^{n}[/tex]

    2. [tex]2\left[sin(\sqrt{x}) - \sqrt{x}cos(\sqrt{x})\right][/tex]

    I think that's right for #2

    3. Still no go
     
  12. Feb 7, 2008 #11
    alrighty brother, i will show you the techniques i learned to solve these puppies.

    Integration by parts w/ borrowing:
    1. Separate the problem into two parts a derivative(x) and an antiderivative(y/dx)
    2. The Derivative must approach zero
    3. Because the antiderivative of Lnx is DUMB, we must always take the derivative(x) of it.

    [tex]\int[/tex]x[tex]^{n}[/tex]lnxdx

    u ___________ dv

    lnx __________ x[tex]^{n}[/tex]

    Continue to take the deriv/antideriv until you are back to where you began ( in this case lnx) and ignore sign changes(+-)

    u ____________ dv

    lnx___________ x[tex]^{n}[/tex]
    1/x___________ x[tex]^{v+1}[/tex]/(n+1)
    lnx___________ Whatever this is

    add each of the parts diagonally varying signs(+/-)

    u ___________dv

    lnx <--------] x[tex]^{n}[/tex]
    1/x <______ ] [+]------> x[tex]^{v+1}[/tex]/(n+1)
    lnx [[-] ________> whatever this is

    lnx+x[tex]^{v+1}[/tex]/(n+1)[/tex] - ( 1/x + whatever this is)


    write them out, distribute, add everything together, cancel out terms and simplify.
     
    Last edited: Feb 7, 2008
  13. Feb 8, 2008 #12
  14. Aug 1, 2010 #13
    i also have a problem in finding the integration of "inetgration of square root of sinx ".....and even "x square divided by 1+ x power 5"
     
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