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Evaluate the integral.

$ \displaystyle \int^{1}_{0} (5x - 5^x) \,dx $

$\frac{5}{2}-\frac{4}{\ln 5}$

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Missouri State University

Campbell University

Harvey Mudd College

Boston College

We're asked to find the integral from 0 to 1 of five X -5 to the X. Power. Now a lot of people have these rules memorized. Otherwise you can google them or you can think about how the derivative of five to the X. Is equal to natural log of five times five to the ex. Well we're going backwards and you better divide. All right, I'll divide by natural log. So that's kind of how I think about the problem. Um So this inter rule. So we have the power rule where we add one to the exponent and then divide by your new exponents. And then again I'm thinking of it backwards. So it's got to be one over natural log of five times five to the X. From 0 to 1. So double check that this is right by taking the derivative of what's in red. It needs to equal what's up here, you know? And if it doesn't then you did something wrong. So now you can plug in your values for X. Your upper bound is one. So one day the second power still wants, we're looking at 5/2ves -1 over natural log of five and five to the first power still five. Now be careful. You do need to actually plug in zero here now zero squared is still zero but 5 to the zero power Is equal to one. So don't get into a habit of leaving that out because then we can simplify this answer because we can combine. I'll stick with the green. This we have five over natural go five or negative and then this will turn into a plus one over natural log five. So we can simplify that. Negative five plus one would be negative four over natural log of five. And this should be a good answer. Just leave it like that and you should be good.