mathschallenge.net

Problem

Solve the integral,

I

=

1 0

[10x] dx

= 9 log(m).

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

Solution

Consider the graph, y = [10^x].

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

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

Evalute,

log(n) 0

[10x] dx.