Problem 63 - Calculus Early Transcendentals

Question

Determine whether the following statements are true and give an explanation or counterexample. a. It is possible that a computer algebra system says \(\int \frac{d x}{x(x-1)}=\ln (x-1)-\ln x\) and a table of integrals says \(\int \frac{d x}{x(x-1)}=\ln \left|\frac{x-1}{x}\right|+C.\) b. A computer algebra system working in symbolic mode could give the result \(\int_{0}^{1} x^{8} d x=\frac{1}{9},\) and a computer algebra system working in approximate (numerical) mode could give the result \(\int_{0}^{1} x^{8} d x=0.11111111.\)

Step-by-Step Solution

Answer

Answer: Yes, the integral expressions from the computer algebra system and table of integrals are the same despite small differences in notation. Also, the definite integral of x^8 from 0 to 1 is approximately equal to 0.111, which confirms the approximate numerical result.

Step 1: Part a: Comparing two expressions of integral

In order to compare these two expressions, we need to simplify the second expression and see if they represent the same function. The first integral expression given is: \(\int \frac{dx}{x(x-1)}=\ln(x-1)-\ln(x)\) The second integral expression given is: \(\int \frac{dx}{x(x-1)}=\ln\left|\frac{x-1}{x}\right|+C\) Now, let's simplify the second expression: \(\ln\left|\frac{x-1}{x}\right| = \ln|x-1|-\ln|x|\) As we are considering the indefinite integral, we can ignore the absolute value signs since they are irrelevant for the constant factor in the general solution. Now, comparing the two expressions, we can see that they are the same since: \(\ln(x-1)-\ln(x) = \ln|x-1|-\ln|x|\) Thus, the answer for Part a is, yes, it is possible for the computer algebra system and table of integrals to show these expressions.

Step 2: Part b: Calculating the definite integral of x^8 and comparing symbolic and numerical results

First, we need to calculate the definite integral of x^8 using symbolic mode: \(\int_{0}^{1} x^8dx\) Using the power rule of integration: \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\) So, the definite integral of x^8 from 0 to 1 is: \(\int_{0}^{1} x^8dx = \left[\frac{x^9}{9}\right]_0^1 = \frac{1^9}{9}-\frac{0^9}{9} = \frac{1}{9}\) Now, let's compare this result to the approximate numerical result given: Symbolic result: \(\int_{0}^{1} x^8 dx = \frac{1}{9} = 0.\overline{1}\) Approximate numerical result: \(\int_{0}^{1} x^8 dx = 0.11111111\) The approximate numerical result is very close to the symbolic result since they both are approximately equal to 0.111. Although there might be some minor differences due to numerical approximation or rounding during the numerical integration process, the results are essentially the same.

Key Concepts

Integral Comparison

Integral comparison is crucial when analyzing different methods or expressions for computing an indefinite integral. An indefinite integral, such as
\[ \int \frac{dx}{x(x-1)} \]
can often be expressed in multiple ways, each mathematically correct. A comparison of two forms, for example, \(\ln(x-1) - \ln(x)\) and \(\ln\left|\frac{x-1}{x}\right| + C\), demonstrates that both expressions represent the same family of functions since the logarithmic properties allow them to be simplified to the same underlying expression. Notably, the presence of an arbitrary constant \(C\) acknowledges the indefinite nature of these integrals. This concept advertises the importance of recognizing and confirming equivalent representations in mathematical analysis and how it can lead to a better understanding of the integral's properties.

Symbolic Integration

Symbolic integration involves finding exact, analytical expressions for integrals using the rules and theorems of calculus. Unlike numerical integration which provides approximate results, symbolic integration aims at delivering expressions in form of algebraic functions, trigonometric functions, logarithms, and others. For instance, \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\) is a result of symbolic integration known as the power rule. It is often performed by computer algebra systems, which can handle complex and tedious integrals with relative ease. Symbolic integration can reveal deeper insights into the mathematical structure and relationships in a problem, which is essential for the theoretical understanding.

Numerical Integration

In contrast to symbolic integration, numerical integration calculates the value of an integral using approximation methods. It is particularly useful when an integral cannot be solved symbolically or when a quick numerical answer is needed. Numerical integration methods include techniques like the Riemann sums, trapezoidal rule, and Simpson's rule. These methods approximate the area under the curve by summing the areas of simple shapes like rectangles or trapezoids. For the integral \(\int_{0}^{1} x^8 dx\), a numerical approximation gave the result 0.11111111, which is an approximation of the exact symbolic result \(\frac{1}{9}\). While numerical integration is indispensable in scientific computing and engineering, it is crucial to bear in mind that it can be prone to errors like truncation and rounding errors.

Logarithmic Functions

Logarithmic functions play a significant role in integral calculus, particularly when working with indefinite integrals. The natural logarithm, denoted as \(\ln(x)\), serves as the inverse of the exponential function \(e^x\), and its properties are often leveraged during integration. For example, the integral \[\int \frac{dx}{x}\] results in \(\ln|x| + C\), exemplifying a logarithmic function's appearance in solutions. Additionally, properties such as \(\ln(ab) = \ln(a) + \ln(b)\) and \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\) are vital for simplifying and comparing logarithmic expressions within integrals. Understanding logarithmic functions is, therefore, essential for solving and manipulating a wide range of integrals.