Name: Problems Plus 17: Integral of x^x
Uploaded: 2023-09-29T05:40:00+00:00
Duration: 38 min 9 s
Description: In this video I go over the integral of x^x from x = 0 to x = 1 and show that it is equal to a power series. Some integrals, such as that for x^x, don't have simple solutions and so we have to resort

2 years ago

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In this video I go over the integral of x^x from x = 0 to x = 1 and show that it is equal to a power series. Some integrals, such as that for x^x, 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 x^x as an exponential function by utilizing log rules involving the natural log, namely converting x^x to e^(x lnx). From here we can modify the power series for e^x, 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 0^0 = 1.

The timestamps of key parts of the video are listed below:

- Problem 17: Integral of x^x 0:00
- Solution: Converting x^x to a power series:37
- Integrating power series term-by-term: 5:18
- Integration by parts: 6:30
- Taking the limit for integral involving natural log: 10:09
- Applying L'Hospital's Rule: 16:46
- Applying Integration by parts again to notice a pattern: 25:12
- Putting the integral all together: 33:54
- Integral of x^x as a series starting from n = 1 (ignore the n = 0 typo): 36:52

This video was taken from my earlier video listed below:

- Infinite Sequences and Series: Problems Plus: https://youtu.be/zjdkQIIdTbg
- HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-problems-plus
- Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FQ96Egr5R7fZGeDTIUKz8P