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Evaluate the integral.

$ \displaystyle \int te^{-3t} dt $

$$

-\frac{1}{3} t e^{-3 t}-\frac{1}{q} e^{-3 t}+C

$$

Integration Techniques

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University of Michigan - Ann Arbor

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Boston College

Hello. So we're trying to evaluate into girl of t E to a native three tt This is a standard integration by parts problem. So you have to figure out a you and a devi to decompose the problem with. Usually you want to find a you that you khun keep differentiating until you get to like one. So that you, Khun, basically get rid of it. And the DV needs to be something that you can integrate visioning your conventional methods. So in this case, it's easy to see the U equals T and D vehicles even Eva three t I say easy because I've done this so many times. It comes with experience. Do you would then be equal to DT? The is equal to e to the negative one third e to the negative three t So what happens next? Put it together. I color coded it. So it's easier to see I put the constants in France. It's easier to work with afterwards, So you have equals to you tempt me which is they won thirty to the in the eternity of three t minus then ago VD you So that's minus ah native one third in a girl of eighteen eighty three T d. T. So you get a plus one third in the next line. Um, and then integrating into ninety three t by itself would just would he also give you another native one third term in a man we get eked native one thirty, eatin a three T minus one ninth eater native three t plus C.