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Math Help - please help- integration

  1. #1
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    please help- integration

    find the integral of
    f (x) = (1+x^2)ArcTan[ x]
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by fair_lady0072002 View Post
    find the integral of
    f (x) = (1+x^2)ArcTan[ x]
    You'll have to be VERY careful of the domain here, but you can try

    x = tan(y)
    dx = sec^2(y) dy

    Then 1 + x^2 --> 1 + tan^2(y) = sec^2(y)

    Your integral then becomes f(y) = sec^4(y)

    -Dan
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  3. #3
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    Quote Originally Posted by fair_lady0072002 View Post
    find the integral of
    f (x) = (1+x^2)ArcTan[ x]
    My, where you do get these complicated integration from?

    Here is what you do,

    INTEGRAL (1+x^2)^2*atan(x)*(1+x^2)^{-1} dx

    Let, u=atan(x) then, u'=(1+x^2)^{-1}
    Then,
    sec^2 u=1-x^2
    cos^2 u=x^2
    Then,
    sec^2 u +2cos^2 u=1+x^2
    Thus,
    INTEGRAL (sec^2 u +2cos^2 u)^2 u*u' dx
    Substitution rule it is composition of outer function,
    INTEGRAL (sec^2 u+2cos^2 u)^2 du
    Open parantheses, (secant*cosine=1)
    INTEGRAL (sec^4 u+4cos^4 u+2) du
    And this is definitely solvable
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    My, where you do get these complicated integration from?

    Here is what you do,

    INTEGRAL (1+x^2)^2*atan(x)*(1+x^2)^{-1} dx

    Let, u=atan(x) then, u'=(1+x^2)^{-1}
    Then,
    sec^2 u=1-x^2
    cos^2 u=x^2
    Then,
    sec^2 u +2cos^2 u=1+x^2
    Thus,
    INTEGRAL (sec^2 u +2cos^2 u)^2 u*u' dx
    Substitution rule it is composition of outer function,
    INTEGRAL (sec^2 u+2cos^2 u)^2 du
    Open parantheses, (secant*cosine=1)
    INTEGRAL (sec^4 u+4cos^4 u+2) du
    And this is definitely solvable
    That's a neat little trick I'll have to remember...

    -Dan
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  5. #5
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    Quote Originally Posted by topsquark View Post
    That's a neat little trick I'll have to remember...

    -Dan
    When you are like me (do not use the dx,dy because their meanings have not been definied) you can only use these tricks to introduce the derivative and hence use substitution rule.

    And thank you for the reputation.
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  6. #6
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    Hello, fair_lady0072002!

    Find the integral of: .(1 + x²) arctan(x) dx
    Let: u= arctan(x) . . x = tan(u) . . dx = sec²(u) du

    . . Then: .1 + x² .= .1 + tan²(x) .= .sec²(x)

    Substitute: . sec²(u) · u · sec²(u) du . = . u·sec^4(u) du

    . . . Good luck!
    Last edited by CaptainBlack; October 17th 2006 at 07:56 PM.
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