For any integer n greater than 1, evaluate this definite integral:
n
∫ ⌊x⌋*x2*{x3} dx
0
Note: ⌊x⌋ is defined as the floor of x, which is equal to the greatest integer less than or equal to x, and:
{x} = x - ⌊x⌋
Numerical integration confirms Steven Lord's analytical solution (not that this was needed).
With delta_x steps of .00001, I calculated the values for n from 1 to 20, put those results into a spreadsheet, made a chart and asked for a trendline, making it a 4th degree polynomial.
The coefficients were 14.9998 instead of 15, 1.9998 instead of 2, and similarly for all the other coefficients.
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Posted by Larry
on 2024-09-27 22:37:44 |
numerical confirmation
