# Integration of a Floor Function

• WhatTheYock
In summary, the integral of floor(ln(floor(1/x))) from 1/4 to 1/2 is equivalent to integrating the function floor(1/x) with values of 2 for 1/3 < x ≤ 1/2 and values of 3 for 1/4 < x ≤ 1/3, resulting in an answer of 1/3. The value of floor(1/x) = 4 at x = 1/4 does not affect the integral.
WhatTheYock
Mod note: Thread moved from the Calculus section to here.
I am having trouble evaluating the integral:

∫floor(ln(floor(1/x)))dx from 1/4 to 1/2

I do not know where or how to start. I probably need a full explanation.

Thanks for any help!

Last edited by a moderator:
What values does floor(1/x) have for 1/4 < x < 1/2?

The values of floor(1/x) go from 2 to 4, then taking logs and then the floor of that would give me 0 to 1, but I still do not understand how to find that area under the curve.

You don't need to worry about floor(1/x) = 4 because it only has that value for x=1/4, and the value of a function at a single point makes no contribution to the integral.

For what values of x do you have floor(1/x) = 2? How about floor(1/x) = 3?

You have x = 1/2 and x = 1/3. Then what?

WhatTheYock, I moved your thread to this homework section. In future posts be sure to use the homework template, and include the problem statement and your efforts.

1 person
As jbunniii said, f(x)= 4 only for x= 1/4 which doesn't affect the integral. For $1/4< x\le 1/3$ $3\le \frac{1}{x}< 4$ so f(x)= 3 and for $1/3< x\le 1/2$ $2\le \frac{1}{x}< 3$ so f(x)= 2.

So $\int_{1/4}^{1/2} f(x) dx= 3(1/3- 1/4)+ 2(1/2-1/3)$.

1 person
@Halls - Note that the problem is to integrate ##\text{floor}(\ln(\text{floor}(1/x))) = \text{floor}(\ln(f(x)))## assuming you are defining ##f(x) = \text{floor}(1/x)##. But the OP should easily be able to make the appropriate modifications.

## What is a floor function?

A floor function, denoted as ⌊x⌋ or floor(x), is a mathematical function that rounds down a decimal number to the nearest integer.

## What is integration?

Integration is a mathematical process that involves finding the area under a curve on a graph. It is the reverse operation of differentiation and is used to solve problems in calculus and other areas of mathematics.

## What is the integration of a floor function?

The integration of a floor function is the process of finding the integral of a function that includes a floor function. It involves evaluating the area under the curve of the function while taking into account the effects of the floor function on the curve.

## What are some common applications of integrating a floor function?

Integrating a floor function is commonly used in fields such as engineering, physics, and economics to solve problems involving discrete quantities or functions with discontinuities. It can also be used to calculate probabilities and in signal processing.

## What are some techniques for integrating a floor function?

Some common techniques for integrating a floor function include using the properties of the floor function, such as its step-like behavior, and breaking the integral into smaller parts. Other techniques may involve using substitution or integration by parts.

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