Problem 64 - Calculus Early Transcendentals

Question

Three different computer algebra systems give the following results: \(\int \frac{d x}{x \sqrt{x^{4}-1}}=\frac{1}{2} \cos ^{-1} \sqrt{x^{-4}}=\frac{1}{2} \cos ^{-1} x^{-2}=\frac{1}{2} \tan ^{-1} \sqrt{x^{4}-1}.\) Explain how they can all be correct.

Step-by-Step Solution

Answer

Short Answer: All three results are correct and equivalent to each other, as shown through simplifications, inverse trigonometric function relationships, and proof of equivalence. They are different forms of the same result, and thus all valid solutions to the given integral.

Step 1: Simplify the given results

For the first result, we can simplify the expression within the function: \(\frac{1}{2} \cos ^{-1}\sqrt{x^{-4}} = \frac{1}{2} \cos ^{-1}\frac{1}{x^{2}}\) This matches the second result, \(\frac{1}{2} \cos ^{-1} x^{-2}\).

Step 2: Identify relationships between the functions

Now, let's denote the third result and establish the relation between its function to the functions of the simplified results: Let \(y = \frac{1}{2} \tan ^{-1}\sqrt{x^{4}-1}\). We can use a well-known relationship between the inverse tangent function and the inverse cosine function: \(\cos ^{-1} z = \frac{\pi}{2} - \tan ^{-1} z\), where \(z\) is any allowed value. In our case, \(z = \sqrt{x^{4}-1}\). Therefore, we can find the corresponding \(y\) in terms of the inverse cosine function: \(y = \frac{1}{2} (\frac{\pi}{2} - \cos ^{-1} \sqrt{x^{4}-1})\)

Step 3: Prove equivalence of the expressions

Now, we will prove that the function \(y\) is equivalent to the simplified result we found in step 1. Compare \(y=\frac{1}{2} (\frac{\pi}{2} - \cos ^{-1} \sqrt{x^{4}-1})\) to the simplified results: \(\frac{1}{2} \cos ^{-1}(\frac{1}{x^{2}}) = \frac{1}{2} \cos ^{-1} x^{-2}\). We can rewrite \(y\) as: \(y = \frac{\pi}{4} - \frac{1}{2} \cos ^{-1} \sqrt{x^{4}-1}\) To show that the expressions are equivalent, we need to demonstrate that the argument inside the inverse cosine function of \(y\) matches the simplified result's argument, i.e., \(\sqrt{x^{4}-1} = \frac{1}{x^{2}}\). Squaring both sides, we obtain: \(x^4 -1 = \frac{1}{x^4}\) Rearranging terms: \(x^8 - x^4 -1=0\) Now, let \(u = x^{4}\), then we get a quadratic equation: \(u^2 -u -1=0\) Here, \(u\) would have two distinct real roots. However, we need only the positive one since \(u = x^4 \ge 0\), as \(x\) is a variable in the domain of the function. We find that \(u = \frac{1 + \sqrt{5}}{2}\) is the positive root, which means: \(x^{4}=\frac{1 + \sqrt{5}}{2}\). Now plugging this back into \(\sqrt{x^{4}-1}\), we get: \(\sqrt{\frac{1 + \sqrt{5}}{2} - 1} = \sqrt{\frac{\sqrt{5}-1}{2}} = \frac{1}{x^{2}}\) This proves that the expressions are equivalent. In conclusion, all three results are actually correct, and they are equivalent to each other, proving that the three computer algebra systems are indeed displaying the same result, just in different forms.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools in calculus, especially when dealing with integrals and trigonometric identities. These functions, which are the inverse of the regular trigonometric functions like sine, cosine, and tangent, work to "undo" these original functions. For example, the inverse sine function denoted as \( \sin^{-1} \), will take a ratio and tell you the angle associated with that ratio, once expressed in radians or degrees.

In the context of definite and indefinite integrals involving complex trigonometric expressions, inverse trigonometric functions like \( \cos^{-1} \) and \( \tan^{-1} \) often make appearances. These can help break down complicated expressions into simpler, more manageable forms. When integrating functions like \( \frac{d x}{x \sqrt{x^{4}-1}} \), spotting potential substitutions involving expressions that resemble inverse trigonometric functions is key to simplifying the process.

• \( \cos^{-1} \) deals with the range of \([-1, 1]\).
• \( \tan^{-1} \) is used to manage angles and slopes.

Through smart algebraic manipulation and the use of known relationships, such as \( \cos^{-1} z = \frac{\pi}{2} - \tan^{-1} z \), we can see how apparently different integral results might be different forms of the same truth.

Computer Algebra Systems

Computer algebra systems (CAS) are powerful software tools designed to perform a variety of mathematical computations, including symbolic algebra manipulation. These systems are essential in modern calculus for automating complex derivations and providing accurate integral solutions.

In the exercise, different CAS outputs might initially look distinct because each system might simplify the integral using different internal algorithms. This results in expressions that appear dissimilar but are, in fact, mathematically equivalent. By understanding the foundational relationships between inverse trigonometric functions and their properties, one can "decode" these expressions to reveal their similarities.

• CAS like Mathematica, Maple, and MATLAB use advanced algorithms to simplify expressions.
• Adding constraints, such as variable ranges, can affect the output form.
• Identifying common patterns and functions, such as \( \tan^{-1} \) and \( \cos^{-1} \), helps in recognizing equivalence.

Ultimately, CAS open the doors to effortlessly handling challenging calculus problems by allowing quick checks and balances, though a deeper understanding of concepts is necessary for interpreting the output meaningfully.

Definite Integrals

Definite integrals compute the area under the curve of a function over a set interval on the x-axis. Unlike indefinite integrals, which provide a family of functions, definite integrals yield a numerical value, representing this area.

In calculus exercises involving functions with inverse trigonometric forms or requiring substitution techniques, manipulating intervals and recognizing expression transformation can ensure that integration results from different systems converge correctly. With the example exercise, even though you're working with indefinite forms, the principles hold. Determining whether those diverse results stem from the same underlying integral function means understanding the appropriateness of any substitutions or transformations used.

• Simplifying integrals helps in identifying when bounds of integration align, if applied.
• Recognizing when transformation and substitution simplify an integral is critical.
• Mathematically proving equivalence of solutions reinforces the correctness of variable bounds.

Thus, scrutinizing the output from various integral computations leads to better insights into understanding how the seemingly complex algebra of definite integrals may reduce through bonded expressions, much like how algebraic identities collapse inverse trigonometric forms in this problem.