Problems

In this question, \(\lfloor x \rfloor\) denotes the greatest integer that is less than or equal to \(x\), so that (for example) \(\lfloor 2.9 \rfloor = 2\), \(\lfloor 2\rfloor = 2\) and \(\lfloor -1.5 \rfloor = -2\). On separate diagrams draw the graphs, for \(-\pi \le x \le \pi\), of:

For any number \(x\), the largest integer less than or equal to \(x\) is denoted by \([x]\). For example, \([3.7]=3\) and \([4]=4\). Sketch the graph of \(y=[x]\) for \(0\le x<5\) and evaluate \[ \int_0^5 [x]\;\d x. \] Sketch the graph of \(y=[\e^{x}]\) for \(0\le x< \ln n\), where \(n\) is an integer, and show that \[ \int_{0}^{\ln n}[\e^{x}]\, \d x =n\ln n - \ln (n!). \]

Show Solution

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Solution:

+ - ↺

\begin{align*} \int_0^5 [x]\;\d x &= 0 \cdot 1 + 1 \cdot 1 + 2 \cdot 1 + 3 \cdot 1 + 4 \cdot 1 \\ &= 10 \end{align*}

+ - ↺

\begin{align*} \int_{0}^{\ln n}[\e^{x}]\, \d x &= \sum_{k=1}^{n-1} \int_{\ln k}^{\ln (k+1)}[\e^{x}]\, \d x \\ &= \sum_{k=1}^{n-1} k \l \ln (k+1) - \ln (k) \r\\ &= \sum_{k=1}^{n-1} \l( (k+1) \l \ln (k+1) - \ln (k) \r - \ln(k+1) \r \\ &= \sum_{k=1}^{n-1} (k+1) \ln (k+1) - \sum_{k=1}^{n-1} k \ln (k) - \sum_{k=1}^{n-1} \ln (k+1) \\ &= n \ln n - 1 \ln 1 - \sum_{k=1}^{n-1} \ln (k+1) \\ &= n \ln n - \ln \l \prod_{k=1}^{n-1} (k+1)\r \\ &= n \ln n - \ln (n!) \end{align*}