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Evaluate the integral.

$ \displaystyle \int x^5 e^{-x^3}\ dx $

$-\frac{e^{-x^{3}}}{3}\left(x^{3}+1\right)+C$

Integration Techniques

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Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

before we evaluate, let's just go ahead and rewrite this. So here I can pull up the X cubed, and then I have X square D x. This might make it easier to see why we should do the use of U equals X cube. Because then do you over three equals X squared DX and then we could rewrite. This is one third of the three is coming from the use up and a rule. This's just you here and then I have eat to the minus you and then this d'You divided by three takes care of this part over here. So now we have a new integral to compute that simple looking than the original. And we can evaluate this using integration by Earth's. So here, let's do dv equals e to the minus you. So v equals either the mine issue don't to get that back the equals minus e to the minus you and then let me use a different letter here w equals you so d w equals. Do you so using our formula for innovation? My parts you can not Usually this is where in in terms of you envy, but you's already being used through the use of this is why I'm using w. So now go ahead and plug in our values for you wouldn't be here. We'Ll have negative you e to the minus you and then we'LL have a double minus here minus here with this one That's a plus in a cruel E to the minus you. So that's minus you eat to the minus you over three and then we integrate will get a minus e to the minus you over three. Now, if this integral the second rule are really the only gonna roll because you difficulty here, you could do a substitution and use of negative you. And with that said, the last step here is to just go back and replace you with X. So we have negative X cubed e to the negative X cubed over three minus e to the negative X cube over three, all divided by three notches toe exponents. And the last thing to do here would be to just pull out that negative deeds of the negative execute over three. And if you do that, you can write, the remaining part is executed the plus one plus e and that's your final answer