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Math Help - Simple antiderrivative

  1. #1
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    Simple antiderrivative

    Mental block on antiderrivating the integral from one to infinity of 1/(square root of x+1).

    Thanks!
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Possible actuary View Post
    Mental block on antiderrivating the integral from one to infinity of 1/(square root of x+1).

    Thanks!
    this is an improper integral, so we have to use limits when evaluating it. for the anti derivative, we use substitution.

    int{0-->infinity} [1/sqrt(x + 1)]dx

    = int{0-->infinity} [(x + 1)^(-1/2)]dx
    let u = x + 1
    => du = dx

    so our integral becomes:

    int{u^(-1/2)}du
    = 2u^(1/2) + C
    = 2(x + 1)^(1/2) + C

    now to evaluate between 0 and infinity, we proceed like this

    int{0-->infinity} [(x + 1)^(-1/2)]dx = lim{N-->infinity} int{0-->N} [(x + 1)^(-1/2)]dx

    = lim{N-->infinity} 2(x + 1)^(1/2) evaluate between N and 0

    = lim{N-->infinity} 2(N + 1)^(1/2) - 2(0 + 1)^(1/2)
    = lim{N-->infinity} 2(N + 1)^(1/2) - 2
    = infinity
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  3. #3
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    Quote Originally Posted by Possible actuary View Post
    Mental block on antiderrivating the integral from one to infinity of 1/(square root of x+1).

    Thanks!
    What Jhevon said, but that it diverges can be deduced by observing that
    1/sqrt(x+1) decreases more slowly than 1/(x+1) the integral over (0,r) of
    which diverges as r -> infty.

    RonL
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    Quote Originally Posted by Possible actuary View Post
    Mental block on antiderrivating the integral from one to infinity of 1/(square root of x+1).

    Thanks!
    INT 1/sqrt(x+1) dx = 2*(x+1)^(1/2)+C

    Now evaluate the integral at limits of 1 and N to get,

    2*(N+1)^(1/2)-2*(1+1)^(1/2)

    Now take N --> oo
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    INT 1/sqrt(x+1) dx = 2*(x+1)^(1/2)+C

    Now evaluate the integral at limits of 1 and N to get,

    2*(N+1)^(1/2)-2*(1+1)^(1/2)

    Now take N --> oo
    Oh yeah, that's right. the limits of integration were from 1 to infinity. i did for 0 to infinity. the answer is pretty much the same though, except for the third to last line, you plug in 1 instead of 0. the answer remains the same, it diverges to infinity.

    = lim{N-->infinity} 2(N + 1)^(1/2) - 2(1 + 1)^(1/2)
    = lim{N-->infinity} 2(N + 1)^(1/2) - 2sqrt(2)
    = infinity

    dang! i've got to learn to read again! i've been misreading posts all week. i guess it's because i'm hardly getting any sleep these days, i'm constantly tired--but maybe that's a sorry excuse
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    Thanks! Could not get the simple integration going.
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  7. #7
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Possible actuary View Post
    Thanks! Could not get the simple integration going.
    that's what we're here for
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