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Problem 26 Medium Difficulty

Use the Table of Integrals on Reference Pages 6-10 to evaluate the integral.

$ \displaystyle \int_0^1 x^4 e^{-x}\ dx $




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Video Transcript

Okay, This question wants us to evaluate this integral. So what we're gonna d'oh is these are integral table to solve this, but that formula could be kind of complicated. So we're just going to use the d ay method using integration by parts. So, in the d column, we just write our polynomial and keep taking derivatives and then all the way down to our final answer before a zero. And then for the eye terms, we just keep integrating those. And now we just puller ladder and what we need another one on this side. So we have this one's plus this one's minus this one's process ones minus this one's plus, and the final one is minus. So that means our anti derivative is next to the fourth e to the minus X. But we have a negative one here, minus for ex cute eat of the minus X minus 12 X squared Eat of the minus X, and you see what's happening here. Every term is a minus, and we're going from 0 to 1 so we can fact you're out our e to the minus X and get a simpler looking anti derivative still going from 0 to 1 and this gets us e to the minus one times negative one minus four minus 12 minus 24 minus 24 plus zero. Sorry plus minus eat of the zero times zero plus zero plus zero plus zero minus 24. So then all we have to do is simplify these because eating zeros one. So we'll just get eat of the negative one times this sum. So 24 minus 24 negative, 48 minus another 12 is negative, 60 minus another five is negative. 65 minus zero is one times negative. 24. So we get our final answer of 24 minus 65 divided by e.

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