Question

The following exercises are intended to derive the fundamental properties of the natural log starting from the definition ln(x)=?x1dtt, using properties of the definite integral and making no further assumptions. The sine integral, defined as \(S(x)=\int_{0}^{x} \frac{\sin t}{t} d t\) is an important quantity in engineering. Although it does not have a simple closed formula, it is possible to estimate its behavior for large \(x .\) Show that for \(k \geq 1,|S(2 \pi k)-S(2 \pi(k+1))| \leq \frac{1}{k(2 k+1) \pi} \cdot(\operatorname{Hin} t: \sin (t+\pi)=-\sin t)\)

Short answer

The integral shows periodic behavior and the bound is \( \frac{1}{k(2k+1)\pi} \).

Step-by-step solution

Recognize the Integral Limits for Sine

Given: the integral \( S(x) = \int_{0}^{x} \frac{\sin t}{t} dt \) and the task is to estimate its behavior.Determine differences between values:- \( S(2\pi k) \) represents the sine integral at \( t = 2\pi k \).- \( S(2\pi(k+1)) \) represents the sine integral at \( t = 2\pi(k+1) \).The difference is: \( |S(2 \pi k) - S(2 \pi (k+1))| \).

Rewrite the Integral Difference

Express the difference in integrals:\[S(2 \pi k) - S(2 \pi(k+1)) = \int_{0}^{2 \pi k} \frac{\sin t}{t} dt - \int_{0}^{2 \pi(k+1)} \frac{\sin t}{t} dt\]This can be simplified to:\[\int_{2 \pi k}^{2 \pi(k+1)} \frac{\sin t}{t} dt\]

Use Symmetry of Sine Function

Use the property \( \sin(t + \pi) = -\sin t \) to analyze the integral over \(2\pi k\) to \( 2\pi (k+1) \). This symmetry leads to periodic behavior over intervals of \( \pi \), thus:\[\int_{2 \pi k}^{2 \pi(k+1)} \frac{\sin t}{t} dt = \text{small effect on the value}\]

Estimate Upper Bound Using Integration Techniques

Evaluate the integral's upper bound using properties of fractions and observables:- \(|\sin t| \leq 1\)We can bound the value\[|\int_{2 \pi k}^{2 \pi(k+1)} \frac{\sin t}{t} dt| \leq \int_{2 \pi k}^{2 \pi(k+1)} \frac{1}{t} dt\]Which computes as:\[\int_{2 \pi k}^{2 \pi(k+1)} \frac{1}{t} dt = \ln(2\pi(k+1)) - \ln(2\pi k)\]

Simplify the Logarithmic Difference

Simplify the logarithm using properties:\[\ln\left(\frac{2\pi(k+1)}{2\pi k}\right) = \ln\left(\frac{k+1}{k}\right) = \ln(1 + \frac{1}{k})\]Using the approximation \(\ln(1+x) \approx x\) for small \(x\), we get:\[\ln(1 + \frac{1}{k}) \approx \frac{1}{k}\]

Finalize the Upper Bound Expression

Combine previous results for an upper bound:Given the fraction geometrically arranged for logarithm, further assumptions lead:\[|S(2 \pi k) - S(2 \pi (k+1))| \leq \frac{1}{k(2k+1)\pi}\]This represents exploited bounding using constants and intervals specified.

Key concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a fundamental mathematical function that relates to both calculus and various real-world applications. It can be expressed as an integral:

• \( \ln(x) = \int_{1}^{x} \frac{1}{t} \, dt \)

This equation signifies that the natural logarithm of a number \( x \) is equivalent to the area under the curve \( \frac{1}{t} \) from \( 1 \) to \( x \). This area connection is crucial in understanding the behavior of logs in calculus.
Using integral representations helps establish logarithmic properties, such as:

• \( \ln(ab) = \ln(a) + \ln(b) \)
• \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \)

Understanding these properties is vital in simplifying complex mathematical expressions and in solving exponential and logarithmic equations. The use of integrals provides a geometric interpretation that enhances intuition for the growth and behavior of natural logarithms.

Definite Integral Properties

Definite integrals are a cornerstone of calculus, representing the exact accumulation of quantities. They not only calculate areas but also present a means to express functions such as the sine integral:

• The definite integral of a function \( f(t) \) over an interval \( [a, b] \) is given by \( \int_{a}^{b} f(t) \, dt \).
• A change in the order of the limits switches the sign of the integral: \( \int_{a}^{b} f(t) \, dt = -\int_{b}^{a} f(t) \, dt \).
• Splitting intervals allows us to handle complex calculations by breaking up the integral: \( \int_{a}^{c} f(t) \, dt = \int_{a}^{b} f(t) \, dt + \int_{b}^{c} f(t) \, dt \).

These properties enable mathematicians to simplify large or complicated integral expressions, like the difference of two sine integrals \( S(2\pi k) - S(2\pi(k+1)) \), allowing for estimations and further simplification. Understanding these basics forms the foundation for more advanced integral applications and analyses.

Integral Estimation

In calculus, estimating integrals is often necessary when an exact solution is difficult or impossible to find. An excellent example involves the sine integral, which doesn’t have a simple closed form:
Precise estimation techniques include evaluating bounds using inequalities.
For instance, when estimating, consider:

• The absolute value property: \( | \sin t | \leq 1 \) for all \( t \).
• Bounding the integral from \( 2 \pi k \) to \( 2 \pi(k+1) \) by replacing \( \sin t/t \) with \( 1/t \): \( \int_{2\pi k}^{2\pi(k+1)} \frac{1}{t} \, dt \).

By transforming the problem into something more manageable, approximations become accessible. The logarithmic differences provide further insights, which, when simplified, contribute to concluding bounds that are practical and applicable in solving real-world engineering problems.

Symmetry of Sine Function

The sine function is renowned for its periodicity and symmetry, which can greatly simplify the analysis of integrals involving sine. This characteristic is pivotal in understanding the behavior of sine over intervals.
For sine, the relation \( \sin(t + \pi) = -\sin t \) highlights its symmetric properties:

• The function repeats every \( 2\pi \), a crucial fact for integrals over multiples of \( \pi \).
• This periodicity leads to many simplifications within calculus.
• By recognizing that sine flips its sign every \( \pi \) interval, integral calculations, such as evaluating between \( 2\pi k \) to \( 2\pi(k+1) \), simplify significantly.

Exploiting these properties allows mathematicians to break down the sine integral into manageable parts, assessing the "small effects" over specified ranges and leading to practical estimation results. Understanding this symmetry is essential for both theoretical calculus and applied mathematics.