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Evaluate the integral.

$ \displaystyle \int^{1}_{0} (x^{10} + 10^x)\,dx $

$\frac{1}{11}+\frac{9}{\ln 10}$

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Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

we're doing the integral from 0 to 1 of X to the 10th. Power Plus 10 to the x power. Yes. Now you can google rules, you memorize rules, you can do a lot of things. But uh if you're familiar with the derivative of a to the X. Is equal to natural log of the base times eight of the X. Well the anti derivative or the integral is working backwards. So it makes sense if you have those rules already memorized divide as you go backwards. So as we're looking at this, we still have the power rule where we add one to the exponent and you multiply by the reciprocal of that. But this is exactly I'm talking about is a lot of students are used to going one way, we'll just go the opposite way. So divide by the natural log of 10 Times 10 to the X. And that's from 0 to 1. So now we're ready for this final answer because we can plug in our upper bound in for both of these excess and uh one to any power still once or do with that. Um And then 10 to the first power just be 10. Now you do need to subtract off plugging in zero and for both of these exits. Now this one is not a big deal because it is zero, But this one is a huge deal because 10 to the zero power is one Tend to the zero power is one. So as we're going to simplify this, We can combine the like terms that are dividing by natural log of 10. Um and it's 10 over natural log of 10 -1 over natural log of 10. So that would be nine over natural log of 10. And I believe this is your best answer. So we can circle it and move on.