Calc Problem Tripping Me Up

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I have this problem for my Calc class and, though I've done it manually and through my calculator, WeBWorK keeps telling me I'm wrong. I hoped someone could tell me if I'm interpreting it incorrectly or maybe just doing the wrong calculation.

A particle that moves along a straight line has velocity

$v(t) = t^2 e^{-2 t}$

meters per second after t seconds. How many meters will it travel during the first t seconds?

To me, this means that I must find

$\int t^2 e^{-2 t} \ dt$

According to everything I've tried, this equals $e^{-2t}(-t^{2}/2-t/2-1/4)$

What am I doing wrong here?

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You did not specify the start and endpoint of the integration. You must integrate from 0 to t. The function you give is a primitive of t²*exp(-2t), but it is not the covered distance at time t.

Let's call this primitive F(t). Then the solution to your problem is F(t) - F(0). Your answer is almost correct, you derived a correct primitive. The answer for the covered distance misses a constant term.

Another way to see that your answer is not complete is to fill in the value t = 0 in your answer. The result also should be zero, but it isn't. At time 0, no distance is covered yet.

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Yes, really you ought to finding the integral

$\int_0^t v(x)dx$

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Of course! I'd actually tried that before, but I had made a mistake integrating at that point. Thanks for the help! This has been bugging me.

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