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Integer Integral

Problem

Solve the integral,

$$\int_{0}^{1} \left [ 10^x \right ]\, dx = 9 - \log (m)$$

Where $\left [ \ \right ]$ is the integer part function and $m$ is an integer to be determined.

Solution

Consider the graph, $y = \left[ 10^x \right]$.


The first step occurs when $10^x = 2\implies x = \log{2} \approx 0.301$, the second step occurs when $10^x = 3 \implies x = \log{3} \approx 0.477$, and so on.

$$\begin{align}I & = 1(\log{2}) + 2(\log{3} - \log{2}) + 3(\log{4} - \log{3}) + \dots + 8(\log{9} - \log{8}) + 9(\log{10} - \log{9}) \\& = 9\log{10} - (\log{2} + \log{3} + \dots + \log{9}) \\& = 9 - \log(9!)\end{align}$$

Evaluate $\displaystyle \int_0^{\log(n)} \left [ 10^x \right ]\, dx$

Problem ID: 148 (Jan 2004)     Difficulty: 4 Star

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