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Math Help - Antiderivative problem

  1. #1
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    Antiderivative problem

    The marginal cost of producing the Xth role of film is given by 5+1/(X+1)^2. The total cost to produce one roll is $1000. find the total cost function.

    to take the antiderivative of 5+1/(X+1)^2 I replaced X+1 with R and wrote

    (5+1)(R^-2). I took the antiderivative of this :

    (5+1)(R^-1)/-1

    I replaced R with X+1 :

    (5+1)/(X+1)

    What am I doing wrong?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Kenneth View Post
    The marginal cost of producing the Xth role of film is given by 5+1/(X+1)^2. The total cost to produce one roll is $1000. find the total cost function.

    to take the antiderivative of 5+1/(X+1)^2 I replaced X+1 with R and wrote

    (5+1)(R^-2). I took the antiderivative of this :

    (5+1)(R^-1)/-1

    I replaced R with X+1 :

    (5+1)/(X+1)

    What am I doing wrong?
    5 + 1/(X+1)^2 = 5 + (x + 1)^-2

    int{5 + (x + 1)^-2}dx = 5x - (x + 1)^-1 + C

    the last step i could do only because the power of x is 1. if we had x^2 + 1 we couldn't do that
    Last edited by Jhevon; March 13th 2007 at 03:29 PM.
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  3. #3
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    Thanks. Could you help me with a similiar problem.

    int{t(t^2+1)^4 + t} dx

    I substitued u for (t^2+1) :

    int{t(u)^4 + t/2t} du
    = int{(u)^4 + t/2} du

    Then I replaced t with (u-1)^1/2 :

    int{(u)^4 + (u-1)^1/2/2} du

    my answer was
    (u^5)/10 + (u^3/2)/3
    = (t^2+1)^5/10 + t^3/3

    is this correct?
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Kenneth View Post
    Thanks. Could you help me with a similiar problem.

    int{t(t^2+1)^4 + t} dx

    I substitued u for (t^2+1) :

    int{t(u)^4 + t/2t} du
    = int{(u)^4 + t/2} du

    Then I replaced t with (u-1)^1/2 :

    int{(u)^4 + (u-1)^1/2/2} du

    my answer was
    (u^5)/10 + (u^3/2)/3
    = (t^2+1)^5/10 + t^3/3

    is this correct?
    half of your answer is right, not sure what happened with the other half. i see you tried to use the substitution on the lone t (which is where you made a mistake), it can work, but it makes life complicated. here's what you do: split the integral into two, and work on each piece separately. (by the way, it should be dt not dx)

    int{t(t^2+1)^4 + t} dt
    = int{t(t^2+1)^4} dt + int{t}dt
    for int{t(t^2+1)^4} dt
    let u = t^2 + 1
    => du = 2t dt
    => (1/2)du = t dt
    so our integral becomes:
    (1/2)int{u^4}du
    = (1/2)(1/5)u^5 + C
    = (1/10)(t^2 + 1)^5 + C

    so int{t(t^2+1)^4} dt + int{t}dt = (1/10)(t^2 + 1)^5 + (1/2)t^2

    for the last term, you had t^3/3, should be (t^2)/2
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