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Homework Help: Velocity, time, distance problem

  1. Jun 9, 2010 #1
    Integration by parts velocity, time, distance problem

    1. The problem statement, all variables and given/known data
    A particle that moves along a straight line has velocity v(t) = t^2*e^(−2t)
    meters per second after t seconds. How many meters will it travel during the first t seconds? This problem comes from an online hw covering sections on Substitution in indefinite integrals and integration by parts

    2. Relevant equations

    3. The attempt at a solution
    It is just the integral right? s(t) = ∫ t^2 e^(-2t) dt

    Let u = t^2 and dv = e^(-2t) dt
    du = 2t dt and v = (-1/2) e^(-2t)

    Integrating by parts, we have s(t) = (-1/2) e^(-2t) • t^2 - (1/2)(2) ∫ t e^(-2t) dt

    Again, integrating by parts:

    Let U = t dV = e^(-2t) dt
    dU = dt V = (-1/2) e^(-2t)

    s(t) = (-1/2) t^2 e^(-2t) - [(-1/2) t e^(-2t) - (-1/2) ∫ e^(-2t) dt] = (-1/2) t^2 e^(-2t) + (1/2) t e^(-2t) + (1/4) e^(-2t) = (-1/2) e^(-2t) ( t^2 - t + 1/2)

    Do I plug in at t=0, s=0 to get my constant. That would give me C=.25, so s(t)= (-1/2)e^(-2t)*(t^2-t+1/2)+.25 but this is not the right answer?
    Last edited: Jun 9, 2010
  2. jcsd
  3. Jun 9, 2010 #2


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    Science Advisor
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    Gold Member

    Re: Integration by parts velocity, time, distance problem

    Wrong sign on the second term. See if that fixes it.
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