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Math Help - Integral of x.arctan(x) with hint with hint

  1. #1
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    Integral of x.arctan(x) with hint with hint

    Hi, I have the problem to find:

    integral of - x.arctan (x)

    the hint / first step is pointed to by first asking to find:

    integral of - x^2 / ( 1 + x^2 )

    I have found the answer to be:

    1/2 ( x^2 +1 ) . arctan (x) - 1/2.x

    I cannot deduce the working to get here though. Any help much appreciated... thanks
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  2. #2
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    We have

     \int x \tan^{-1}(x)~dx

     = \frac{1}{2} \int \tan^{-1}(x) ~d(x^2+1)

    We now use integration by parts ,

     u = x^2 + 1 ~,~ v = \tan^{-1}(x)

     = \frac{1}{2} \int v ~du

     = \frac{1}{2} [ (u)v - \int u dv ]

     = \frac{1}{2} [   (x^2 + 1 )  \tan^{-1}(x) - \int (x^2 + 1  )d( \tan^{-1}(x)) ]

     = \frac{1}{2} [  (x^2 + 1 ) \tan^{-1}(x)  - \int (x^2 + 1 ) \frac{dx}{x^2+1}]

     = \frac{1}{2} [ (x^2 + 1 ) \tan^{-1}(x)  - \int dx ]

     = \frac{1}{2} [  (x^2 + 1 )  \tan^{-1}(x) - x ] + C
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  3. #3
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    I'm sorry, your solution is very good and most succinct but to gain full marks for my assignment think i have to use the integral of
    x^2 / ( 1+x^2 )

    I have, from more directly integrating by parts of x.arctan (x) where

    u=arctan(x) giving du= dx / (1+x^2)

    dv=x giving v=1/2.x^2

    in formula int u.dv = u.v. - int v.du

    shows the significance of the x^2 / ( 1+x^2 ) term by -

    1/2.x^2.arctan(x) - 1/2. int x^2 / ( 1+x^2 )

    I am glad to have seen your method, I have not seen that technique before (your very first step of sub know integral of x^2 + 1, does it have a name?) I have tried to see how it might be relavent to my current position as I have attempted to explain - but failed again unfortunately. If you could again assist i would be most humbled. Also, are you inserting Images made in / with word equation editor?

    many thanks !
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  4. #4
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    Quote Originally Posted by Hytholoday71 View Post
    I'm sorry, your solution is very good and most succinct but to gain full marks for my assignment think i have to use the integral of
    x^2 / ( 1+x^2 )

    I have, from more directly integrating by parts of x.arctan (x) where

    u=arctan(x) giving du= dx / (1+x^2)

    dv=x giving v=1/2.x^2

    in formula int u.dv = u.v. - int v.du

    shows the significance of the x^2 / ( 1+x^2 ) term by -

    1/2.x^2.arctan(x) - 1/2. int x^2 / ( 1+x^2 )

    I am glad to have seen your method, I have not seen that technique before (your very first step of sub know integral of x^2 + 1, does it have a name?) I have tried to see how it might be relavent to my current position as I have attempted to explain - but failed again unfortunately. If you could again assist i would be most humbled. Also, are you inserting Images made in / with word equation editor?

    many thanks !
    MHF policy is to not knowingly help with questions that count towards a student's final grade. It's meant to be the work of the student, not the work of other people.

    Thread closed.
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