Although this without complex analysis approach is extremely cool, the most elegant approach award goes to basic complex analysis (without any contour integration!):
https://youtu.be/zil6BXW6yhE
For the Leibniz rule approach:
https://youtu.be/duZudUtyTaE
And for those awesome two equal integrals towards the end:
https://youtu.be/jQ1t5_ZgvnI