The Ultimate Calculus Proof: This Integral Calculates e to 45 Decimal Places

Added: 2026-05-21

[00:00:00]Want to calculate Euler's number to an astonishing 45 decimal places?

[00:00:04]Here's the ultimate finale to our calculus proof series pushing our integral to the 16th power.

[00:00:10]First, expand the polynomial in the integrand into a 32nd degree polynomial.

[00:00:15]Integrating it with e to the x requires applying integration by parts 32 times.

[00:00:21]When everything simplifies, the exact area becomes a colossal 33-digit number times e minus another 33-digit constant.

[00:00:31]Now, look at the graph.

[00:00:33]The peak is almost invisible, but the curve never dips below the x-axis. So, the total area must be strictly positive.

[00:00:40]Solving that inequality for e gives us an incredibly precise fraction. And when we check the accuracy, the result is stunning. That single fraction matches the true value of e to 45 decimal places.