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The sales of a plastic widget were estimated to be:

$P(t)  =  8000te ^{−0.5t}$

where $t$ is in weeks, and $P(t)$ is in units per week.

How many widgets were sold in the first $9$ weeks?

To start, I need to definite integral. This means that I need a $v$, $\mathrm{d}v$, $u$, and $\mathrm{d}u$.

I know $u= t$, $du = dt$, $dv = e^{-0.5t}$ and $v= -e^{-0.5t}/{0.5}$

So then it should be

$\int^9_08000te^{-.5t}dt$

After that I should switch it to

$8000\int^9_0te^{-.5t}dt$

But I'm not sure what to do after that point to get the answer.

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    Perhaps a more informative title would be helpful. – 2017-02-13
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    definite integral word problem? is that better? – 2017-02-13
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    If $u=t$, why is $du = 1\, dx$? And, what happens to $du$? Also, is the exponent $-0.5t$ or $-0.05t$? (You switch from one to the other half way through your post.) If you pull out the constant $8000$ from the integral, why do you still have it under the integral sign? – 2017-02-13
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    made the edits. – 2017-02-13
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    Good. You made most of them. You still miss a $dt$ in your integrals. Next, you can proceed with Petru's suggestion. – 2017-02-13

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In order to solve the integral, taking into account you are using parts method:

$uv\int vu' dt$

Solve that indefinite integral (comment if you need help there), compute and then apply the boundaries

And of course, the 8000 must be outside the integral, not duplicated inside.

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    I need a double check on the intergral, should the answer end up 3.7556? Then I should multiple it by 8000 to get 30045 (rounded to nearest whole number)? – 2017-02-13
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    Yep, that's the answer. – 2017-02-13