Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. It is possible for a computer algebra system to give the result \(\int \frac{d x}{x(x-1)}=\ln (x-1)-\ln x\) and a table of integrals to give the result \(\int \frac{d x}{x(x-1)}=\ln \left|\frac{x-1}{x}\right|+C\) 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 \(\int_{0}^{1} x^{8} d x=0.11111111\).

Short answer

Based on the analysis, the correct expression for the integral of \(\frac{1}{x(x-1)}\) is \(\ln\frac{|x|}{|x-1|}+C\). The difference between the expressions given in the problem is due to the fact that the CAS output does not include the absolute values, making it less general. For the definite integral \(\int_{0}^{1} x^8 dx\), both the symbolic and approximate outputs from the CAS, \(\frac{1}{9}\) and \(0.11111111\), are reasonable results. The small difference between the values is due to the finite numerical precision when using approximate mode in the computer algebra system.

Step-by-step solution

a. Comparing the two integral expressions: \(\int \frac{d x}{x(x-1)}\)

To verify the correctness of both expressions for the integral, let's first integrate the given function and compare the results: 1. Rewrite the function as a sum of partial fractions: $$\frac{1}{x(x - 1)} = \frac{A}{x} + \frac{B}{x - 1}$$ Solve for A and B: $$A(x - 1) + Bx = 1$$ Substitute \(x = 1\) to get \(A = 1\). Substitute \(x = 0\) to get \(B = -1\). Therefore: $$\frac{1}{x(x - 1)} = \frac{1}{x} - \frac{1}{x - 1}$$ 2. Integrate separate terms: $$\int \frac{d x}{x(x-1)} = \int \left(\frac{1}{x} - \frac{1}{x - 1} \right) d x = \int \frac{1}{x} d x - \int \frac{1}{x - 1} d x$$ Applying the integral of \(\int\frac{1}{x}dx = \ln|x|\): $$\int \frac{d x}{x(x-1)} = \ln |x| - \ln |x - 1| = \ln \frac{|x|}{|x-1|}$$ The result matches the entry from the table of integrals: \(\ln \left|\frac{x-1}{x}\right|+C\). The output of the computer algebra system (\(\ln (x-1)-\ln x\)) is equivalent to our result when \(x>1\). For the other values of \(x\), it fails to include the absolute values. Thus, the table of integrals provides a more general result.

b. Analyzing CAS output for \(\int_{0}^{1} x^{8} d x\)

We need to check if the outputs \(\frac{1}{9}\) and \(0.11111111\) are reasonable results for the definite integral \(\int_{0}^{1} x^{8} d x\). 1. Integrate the function symbolically: We will find the antiderivative using the Power Rule: \(\int x^n d x = \frac{x^{n+1}}{n+1}+C\). $$\int x^{8} d x = \frac{x^{9}}{9}+C$$ Now, we will compute the definite integral: $$\int_{0}^{1} x^{8} d x = \frac{1^{9}}{9} - \frac{0^{9}}{9} = \frac{1}{9}$$ 2. Analyze the approximate result: The output from the CAS in approximate mode for the integral is \(0.11111111\). Comparing this with the symbolic result \(\frac{1}{9}\), we can observe that: $$0.11111111 = \frac{1}{9} \approx 0.111111 \bar{1}$$ The error in the approximation is very small, and the approximate result is consistent with the symbolically computed result. The difference in values can be attributed to finite numerical precision when using approximate mode in the CAS. In conclusion, both the symbolic and approximate modes of CAS provide reasonable results for the integral.

Key concepts

Computer Algebra System

A Computer Algebra System (CAS) is a software tool that allows users to perform a variety of mathematical calculations including algebraic, trigonometric, calculus problems, and more complex mathematics. The beauty of a CAS lies in its ability to manipulate mathematical expressions in symbolic form.

This includes simplification, factorization, solving equations, and importantly for our context, integrals and derivatives. When working with definite integrals, as in the case of \(\int_{0}^{1} x^{8} dx\), a CAS can provide an exact symbolic result, like \(\frac{1}{9}\), or an approximate numeric result, such as \(0.11111111\). While symbolic results give the precise mathematical expression, numerical results provide a decimal approximation, which can be especially useful for evaluating integrals that do not have a closed-form symbolic answer or for quick estimates when high precision is not required.

The strength of CAS is particularly visible when dealing with complex or tedious analytical calculations that would be time-consuming to perform by hand.

Symbolic Integration

Symbolic integration is a process used to find the antiderivative or integral of a mathematical expression in terms of symbols rather than numerical values. This method provides an exact answer and generally includes variables and constants.

For example, integrating the function \(x^{8}\) over a definite range, we apply the Power Rule of integration symbolically to obtain \(\frac{x^{9}}{9}+C\). In definite integration, we evaluate this antiderivative at the upper and lower bounds of integration to find the precise area under the curve. This type of integration is essential in scenarios where the purity of the mathematical form is required, and it contributes significantly to understanding the underlying principles behind the integral.

Partial Fractions Decomposition

The technique of Partial Fractions Decomposition is an effective way to integrate rational functions, which are quotients of polynomials. By breaking down a complex fraction into simpler parts, each component can be integrated individually using more straightforward methods.

For instance, in the solution above, \(\frac{1}{x(x-1)}\) is decomposed into \(\frac{A}{x} + \frac{B}{x-1}\) by solving for the constants A and B. This decomposition allows for the integral to be split into two separate, easier-to-integrate logs, \(\ln |x|\) and \(\ln |x-1|\). Understanding this process is crucial for students as it equips them with a method to tackle integrals of rational functions that would otherwise be difficult to solve directly.

Power Rule Integration

One of the most fundamental techniques in calculus is the Power Rule for Integration, which simplifies the process of finding antiderivatives of polynomials. The rule states that the integral of \(x^n\) with respect to x is \(\frac{x^{n+1}}{n+1}\) plus the constant of integration C, where n is any real number other than -1.

This rule is applied to the function \(x^{8}\) to obtain its antiderivative \(\frac{x^{9}}{9}+C\), and by evaluating this at the bounds of the definite integral from 0 to 1, we find \(\frac{1}{9}\). It is essential for students to master this rule for swift and correct integration of polynomials. Whether in symbolic computation or manual calculation, the Power Rule is a cornerstone of integral calculus.