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Sample Test 4 Question 3

Let \(a>1\) be an integer. Give a non-integral expression in terms of \(a\) for \(F(a)=\displaystyle\int_1^a (-1)^{\lfloor x \rfloor} \lfloor x \rfloor^{-1} dx,\) where \(\lfloor x\rfloor\) is the greatest integer less than or equal to \(x.\)

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Hints

  • Hint 1
    Sketch the graph \(y = \frac{1}{\lfloor{x}\rfloor}\) for \(x > 1.\)
  • Hint 2
    How does multiplying it by \((-1)^{\lfloor{x}\rfloor}\) change the graph?
  • Hint 3
    An integral of a function is just the area under its graph.
  • Hint 4
    Could you express the total area as a sum?

Solution

Plotting the function \(y = (-1)^{\lfloor{x}\rfloor}\frac{1}{\lfloor{x}\rfloor}\) yields rectangles of width \(1\) and height \(-1, \frac{1}{2}, -\frac{1}{3}, \frac{1}{4}, -\frac{1}{5}, \ldots.\) Since an integral of a function is just the area under its graph, \(F(a)=\sum_{i=1}^{a-1}\frac{(-1)^i}{i}.\)

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