Solving $\left| \int_{\pi}^{\frac{3 \pi}{2}} \frac{ \sin X}{x} Dx \right| < \frac{n}{10}$ By Hand

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Introduction

In this article, we will delve into the world of real analysis and integration, tackling a challenging problem that involves finding the smallest natural number $n$ such that a given integral inequality holds true. The problem, which originated from a now-deleted MSE post, requires us to evaluate the absolute value of a definite integral and compare it to a fraction involving $n$. Our goal is to solve this inequality by hand, using mathematical techniques and reasoning to arrive at the correct solution.

Understanding the Problem

The problem statement is as follows:

$$\left| \int_{\pi}^{\frac{3 \pi}{2}} \frac{ \sin x}{x} dx \right| < \frac{n}{10}$$

Here, we are dealing with a definite integral of the function $\frac{\sin x}{x}$, evaluated from $\pi$ to $\frac{3\pi}{2}$. The absolute value of this integral is then compared to the fraction $\frac{n}{10}$, where $n$ is a natural number. Our task is to find the smallest value of $n$ that satisfies this inequality.

Analyzing the Integral

To begin solving this problem, let's first analyze the integral itself. The function $\frac{\sin x}{x}$ is a well-known function in mathematics, and its behavior is quite interesting. As $x$ approaches $0$, the function approaches $1$, but as $x$ increases, the function oscillates between positive and negative values. This oscillatory behavior is due to the presence of the sine function in the numerator.

Using the Squeeze Theorem

One of the key techniques we can use to solve this problem is the squeeze theorem. This theorem states that if we have three functions $f(x)$, $g(x)$, and $h(x)$, such that $f(x) \leq g(x) \leq h(x)$ for all $x$ in a given interval, then if $\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L$, we have $\lim_{x \to a} g(x) = L$.

In our case, we can use the squeeze theorem to bound the integral from above and below. We know that $|\sin x| \leq 1$ for all $x$, so we can write:

$$\left| \int_{\pi}^{\frac{3 \pi}{2}} \frac{ \sin x}{x} dx \right| \leq \int_{\pi}^{\frac{3 \pi}{2}} \frac{1}{x} dx$$

This gives us an upper bound for the integral, which we can evaluate using basic integration techniques.

Evaluating the Upper Bound

To evaluate the upper bound, we can use the fact that the integral of $\frac{1}{x}$ is $\ln|x|$. Therefore, we have:

$$\int_{\pi}^{\frac{3 \pi}{2}} \frac{1}{x} dx = \ln\left|\frac{3\pi}{2}\right| - \ln|\pi|$$

Simplifying this expression, we get:

$$\int_{\pi}^{\frac{3 \pi}{2}} \frac{1}{x} dx = \ln\left|\frac{3\pi}{2\pi}\right| = \ln\left|\frac{3}{2}\right|$$

Using the Lower Bound

To find a lower bound for the integral, we can use the fact that $|\sin x| \geq 0$ for all $x$. Therefore, we can write:

$$\left| \int_{\pi}^{\frac{3 \pi}{2}} \frac{ \sin x}{x} dx \right| \geq 0$$

This gives us a lower bound for the integral, which is $0$.

Combining the Bounds

Now that we have both an upper and lower bound for the integral, we can combine them to get a final bound. Using the squeeze theorem, we have:

$$0 \leq \left| \int_{\pi}^{\frac{3 \pi}{2}} \frac{ \sin x}{x} dx \right| \leq \ln\left|\frac{3}{2}\right|$$

Finding the Smallest $n$

Finally, we can use this bound to find the smallest value of $n$ that satisfies the inequality. We want to find the smallest $n$ such that:

$$\left| \int_{\pi}^{\frac{3 \pi}{2}} \frac{ \sin x}{x} dx \right| < \frac{n}{10}$$

Using the upper bound we found earlier, we can write:

$$\ln\left|\frac{3}{2}\right| < \frac{n}{10}$$

Solving for $n$, we get:

$$n > \frac{10\ln\left|\frac{3}{2}\right|}{1}$$

Using a calculator to evaluate the right-hand side, we get:

$$n > 1.83$$

Since $n$ must be a natural number, the smallest value of $n$ that satisfies the inequality is $n = 2$.

Conclusion

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Q: What is the problem statement?

A: The problem statement is to find the smallest natural number $n$ such that the absolute value of the definite integral of $\frac{\sin x}{x}$ from $\pi$ to $\frac{3\pi}{2}$ is less than $\frac{n}{10}$.

Q: Why is this problem important?

A: This problem is important because it requires the application of mathematical techniques and reasoning to solve a challenging inequality. It also provides a good example of how to use the squeeze theorem to bound an integral from above and below.

Q: What is the squeeze theorem?

A: The squeeze theorem is a mathematical technique that states that if we have three functions $f(x)$, $g(x)$, and $h(x)$, such that $f(x) \leq g(x) \leq h(x)$ for all $x$ in a given interval, then if $\lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L$, we have $\lim_{x \to a} g(x) = L$.

Q: How did you use the squeeze theorem to solve this problem?

A: We used the squeeze theorem to bound the integral from above and below. We first found an upper bound for the integral by using the fact that $|\sin x| \leq 1$ for all $x$. We then found a lower bound for the integral by using the fact that $|\sin x| \geq 0$ for all $x$. Finally, we combined the upper and lower bounds to get a final bound for the integral.

Q: What was the final bound for the integral?

A: The final bound for the integral was $0 \leq \left| \int_{\pi}^{\frac{3 \pi}{2}} \frac{ \sin x}{x} dx \right| \leq \ln\left|\frac{3}{2}\right|$.

Q: How did you find the smallest value of $n$ that satisfies the inequality?

A: We used the final bound for the integral to find the smallest value of $n$ that satisfies the inequality. We wanted to find the smallest $n$ such that $\left| \int_{\pi}^{\frac{3 \pi}{2}} \frac{ \sin x}{x} dx \right| < \frac{n}{10}$. Using the upper bound we found earlier, we can write $\ln\left|\frac{3}{2}\right| < \frac{n}{10}$. Solving for $n$, we get $n > \frac{10\ln\left|\frac{3}{2}\right|}{1}$. Using a calculator to evaluate the right-hand side, we get $n > 1.83$. Since $n$ must be a natural number, the smallest value of $n$ that satisfies the inequality is $n = 2$.

Q: What is the final answer?

A: The final answer is $n = 2$.

Q: What are some common mistakes to avoid when solving this problem?

A: Some common mistakes to avoid when solving this problem include:

• Not using the squeeze theorem to bound the integral from above and below.
• Not finding a lower bound for the integral.
• Not combining the upper and lower bounds to get a final bound for the integral.
• Not using a calculator to evaluate the right-hand side of the inequality.

Q: What are some tips for solving this problem?

A: Some tips for solving this problem include:

• Make sure to use the squeeze theorem to bound the integral from above and below.
• Find a lower bound for the integral.
• Combine the upper and lower bounds to get a final bound for the integral.
• Use a calculator to evaluate the right-hand side of the inequality.
• Check your work carefully to avoid common mistakes.