Evaluate $\int_1^n \frac{\{ T \}^2-\{ T \}+1/6}{t^3}\,dt$

Introduction

In this article, we will delve into the evaluation of a definite integral involving a fractional part function. The integral in question is $\displaystyle\int_1^n \frac{\{ t \}^2-\{ t \}+1/6}{t^3}\,dt$, where $\{t\}$ denotes the fractional part of $t$. This type of integral is a classic example of a problem that requires a deep understanding of mathematical analysis and the properties of fractional part functions.

Understanding the Fractional Part Function

Before we proceed with the evaluation of the integral, let's take a moment to understand the fractional part function. The fractional part function $\{t\}$ is defined as the decimal part of a real number $t$. In other words, if $t$ is a real number, then $\{t\}$ is the number obtained by subtracting the greatest integer less than or equal to $t$ from $t$. For example, if $t = 3.7$, then $\{t\} = 0.7$.

Properties of the Fractional Part Function

The fractional part function has several important properties that we will need to use in the evaluation of the integral. One of the most important properties is that the fractional part function is periodic with period $1$. This means that for any real number $t$, we have $\{t+1\} = \{t\}$. Another important property is that the fractional part function is continuous and differentiable on the interval $[0,1)$.

Evaluating the Integral

Now that we have a good understanding of the fractional part function, let's proceed with the evaluation of the integral. To do this, we will use the following approach:

1. Split the integral into two parts: We will split the integral into two parts, one from $1$ to $n-1$ and another from $n-1$ to $n$.
2. Use the periodicity of the fractional part function: We will use the periodicity of the fractional part function to simplify the integral.
3. Use the continuity and differentiability of the fractional part function: We will use the continuity and differentiability of the fractional part function to evaluate the integral.

Splitting the Integral

Let's start by splitting the integral into two parts:

$$\int_1^n \frac{\{ t \}^2-\{ t \}+1/6}{t^3}\,dt = \int_1^{n-1} \frac{\{ t \}^2-\{ t \}+1/6}{t^3}\,dt + \int_{n-1}^n \frac{\{ t \}^2-\{ t \}+1/6}{t^3}\,dt$$

Using the Periodicity of the Fractional Part Function

Now, let's use the periodicity of the fractional part function to simplify the first integral:

$$\int_1^{n-1} \frac{\{ t \}^2-\{ t \}+1/6}{t^3}\,dt = \int_0^{n-2} \frac{\{ t+1 \}^2-\{ t+1 \}+1/6}{(t+1)^3}\,dt$$

Using the periodicity of the fractional part function, we can rewrite the integral as:

$$\int_0^{n-2} \frac{\{ t+1 \}^2-\{ t+1 \}+1/6}{(t+1)^3}\,dt = \int_0^{n-2} \frac{\{ t \}^2-\{ t \}+1/6}{(t+1)^3}\,dt$$

Using the Continuity and Differentiability of the Fractional Part Function

Now, let's use the continuity and differentiability of the fractional part function to evaluate the second integral:

$$\int_{n-1}^n \frac{\{ t \}^2-\{ t \}+1/6}{t^3}\,dt = \int_{n-1}^n \frac{\{ t \}^2-\{ t \}+1/6}{t^3}\,dt$$

Using the continuity and differentiability of the fractional part function, we can rewrite the integral as:

$$\int_{n-1}^n \frac{\{ t \}^2-\{ t \}+1/6}{t^3}\,dt = \frac{1}{n^3} \int_{n-1}^n \frac{\{ t \}^2-\{ t \}+1/6}{1}\,dt$$

Evaluating the Integral

Now that we have simplified the integral, let's evaluate it:

$$\int_1^n \frac{\{ t \}^2-\{ t \}+1/6}{t^3}\,dt = \int_0^{n-2} \frac{\{ t \}^2-\{ t \}+1/6}{(t+1)^3}\,dt + \frac{1}{n^3} \int_{n-1}^n \frac{\{ t \}^2-\{ t \}+1/6}{1}\,dt$$

Using the properties of the fractional part function, we can rewrite the integral as:

$$\int_0^{n-2} \frac{\{ t \}^2-\{ t \}+1/6}{(t+1)^3}\,dt = \int_0^{n-2} \frac{t^2-t+1/6}{(t+1)^3}\,dt$$

Evaluating the integral, we get:

$$\int_0^{n-2} \frac{t^2-t+1/6}{(t+1)^3}\,dt = \frac{1}{2} \ln \left( \frac{n-2}{n-1} \right) + \frac{1}{2} \ln \left( \frac{n-1}{n} \right)$$

Conclusion

In this article, we have evaluated the definite integral $\displaystyle\int_1^n \frac{\{ t \}^2-\{ t \}+1/6}{t^3}\,dt$, where $\{t\}$ denotes the fractional part of $t$. We have used the properties of the fractional part function, including its periodicity and continuity, to simplify the integral and evaluate it. The final answer is:

$$\int_1^n \frac{\{ t \}^2-\{ t \}+1/6}{t^3}\,dt = \frac{1}{2} \ln \left( \frac{n-2}{n-1} \right) + \frac{1}{2} \ln \left( \frac{n-1}{n} \right)$$

Introduction

In our previous article, we evaluated the definite integral $\displaystyle\int_1^n \frac{\{ t \}^2-\{ t \}+1/6}{t^3}\,dt$, where $\{t\}$ denotes the fractional part of $t$. In this article, we will answer some of the most frequently asked questions about this integral and provide additional insights into its properties and applications.

Q: What is the fractional part function?

A: The fractional part function $\{t\}$ is defined as the decimal part of a real number $t$. In other words, if $t$ is a real number, then $\{t\}$ is the number obtained by subtracting the greatest integer less than or equal to $t$ from $t$. For example, if $t = 3.7$, then $\{t\} = 0.7$.

Q: What are the properties of the fractional part function?

A: The fractional part function has several important properties, including:

• Periodicity: The fractional part function is periodic with period $1$. This means that for any real number $t$, we have $\{t+1\} = \{t\}$.
• Continuity: The fractional part function is continuous on the interval $[0,1)$.
• Differentiability: The fractional part function is differentiable on the interval $[0,1)$.

Q: How did you evaluate the integral?

A: We evaluated the integral by splitting it into two parts, one from $1$ to $n-1$ and another from $n-1$ to $n$. We then used the periodicity of the fractional part function to simplify the first integral and the continuity and differentiability of the fractional part function to evaluate the second integral.

Q: What is the final answer to the integral?

A: The final answer to the integral is:

$$\int_1^n \frac{\{ t \}^2-\{ t \}+1/6}{t^3}\,dt = \frac{1}{2} \ln \left( \frac{n-2}{n-1} \right) + \frac{1}{2} \ln \left( \frac{n-1}{n} \right)$$

Q: What are the applications of this integral?

A: This integral has several applications in mathematical analysis, including:

• Approximation of functions: The integral can be used to approximate the value of a function at a given point.
• Numerical integration: The integral can be used to approximate the value of a definite integral.
• Probability theory: The integral can be used to model probability distributions.

Q: How can I use this integral in my own work?

A: You can use this integral in your own work by applying the techniques and methods used in its evaluation. For example, you can use the periodicity of the fractional part function to simplify integrals involving periodic functions, or you can use the continuity and differentiability of the fractional part function to evaluate integrals involving continuous and differentiable functions.

Conclusion

In this article, we have answered some of the most frequently asked questions about the definite integral $\displaystyle\int_1^n \frac{\{ t \}^2-\{ t \}+1/6}{t^3}\,dt$ and provided additional insights into its properties and applications. We hope that this article has been helpful in understanding the evaluation of this integral and its potential applications in mathematical analysis.