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Problem 14 Medium Difficulty

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

$ \displaystyle \int_0^{\pi} x^3 \sin x\ dx $


$\pi^{3}-6 \pi$


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Video Transcript

Okay, This question asks us to compete the value of this integral. So one way you could do it is with a table, as they suggest. But since that involves a lot of Riker jh in, it might actually be easier to use integration by parts because we know how to do this. So we'll just use the d I method or the ladder method, whatever you want to say. So we have our d column here and there. I call him here. So our first term is X cubed, and then we just keep taking derivatives. Then for I column, we start with sine X and then we integrate negative coastline X negative sign X co sign X sine X and then we multiply the diagonals together, remembering that it goes plus minus plus minus. So we get negative. X Cube co sign X minus a minus. So plus three x squared Signed X Plus six ex co sign X minus six signings and we're evaluating this from zero to pie. So if we plug in pie, we get negative pi cubed times negative one plus zero plus six pie times negative one plus zero minus. So this is our upper limit and then our lower limit is well X cube zero x squared to zero access zero and sign X zero. So zero. So our final answer turns out to be just negative. Pi cubed times in negative one plus six pie times negative one, which is hi cubed minus six pie And that's our final answer. And again, I just used the D ay method because, quite frankly, I think it's much faster than plugging into the table over and over again. But you're more than welcome to do that if the D ay method does not appeal to you.

University of Michigan - Ann Arbor
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