The CSAT and Practice[+] are designed by the Climb Foundation to help candidates. We are advocates for more opportunity to shine and less opportunity to fail, and we strive to level the playing field. 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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