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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