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Math Help - integration by substitution

  1. #1
    Newbie
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    Exclamation integration by substitution

    please help me with this, i cannot seem to integrate these two...

    integrate

    dx/ x^4(1+x^2)^(1/2)

    using the substitution x = 1/ (u^2-1)^(1/2)

    and

    x^2(1+x^2)^(1/2) using the substitution x = tan u

    p.s.

    does anyone know exactly how to write these equations and formulas "properly", with the integral sign and such? I have a program called mathematica but it is not compatible here on the forum.
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  2. #2
    Super Member

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    Hello, mathshelpneeded!

    I can get you started on the second one . . .


    x▓(1 + x▓)^Ż dx, .using the substitution: x = tanθ

    We have: .x = tanθ . . dx = sec▓θĚdθ
    . . and the radical becomes: secθ

    Substitute: .
    tan▓θĚsecθĚsec▓θĚdθ . = . sec│θĚtan▓θĚdθ

    This integral is particularly annoying to integrate.
    . . It has an odd-powered secant and an even-powered tangent.
    It requires integration by parts and some fancy algebra.

    I'll jump to the punchline:
    . . (1/8)
    [2Ěsec│θĚtanθ + 3ĚsecθĚtanθ + 3Ěln|secθ + tanθ|] + C

    and let you back-substitute . . .

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