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Integer Integral

Problem

Solve the integral,

I =
1


0
[10x] dx = 9 minus log(m).

Where [ ] is the integer part function and m is an integer to be determined.


Solution

Consider the graph, y = [10x].


The first step occurs when 10x = 2 implies x = log2 approximately 0.301, the second step occurs when 10x = 3 implies x = log3 implies 0.477, and so on.

I = 1(log2) + 2(log3 minus log2) + 3(log4 minus log3) + ... + 9(log10 minus log9)
  = 9log10 minus (log2 + log3 + ... + log 8)
  = 9 minus log(8!)

Evalute,
log(n)


0
[10x] dx.
Problem ID: 148 (Jan 2004)     Difficulty: 4 Star

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