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In this video I go over the integral of xx from x = 0 to x = 1 and show that it is equal to a power series. Some integrals, such as that for xx, don't have simple solutions and so we have to resort to writing them out in power series; in fact this is how calculators solve most integrals! I start off by rewriting xx as an exponential function by utilizing log rules involving the natural log, namely converting xx to ex ln x. From here we can modify the power series for ex, and after several integrals using term-by-term integration, as well as integration by parts, we can notice a pattern to finally solve the integral. The result is now a basic power series. Truly amazing stuff! Note that in this integration, I defined 00 = 1.
The timestamps of key parts of the video are listed below:
This video was taken from my earlier video listed below:
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