## Integer Integral

#### Problem

Solve the integral,

I | = |
| [10^{x}] 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, |
| [10^{x}] dx. |

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