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.

Short answer

Question: Explain how the three expressions given by the computer algebra systems can be correct solutions for the integral: $$\int \frac{dx}{x\sqrt{x^4 - 1}}$$ Answer: The three expressions were simplified, and it was shown that the first two expressions are the same after simplification. Then, the relationship between the remaining expressions was analyzed using trigonometric properties, specifically the property that states $$\cos^{-1} y = \tan^{-1}\sqrt{1 - y^2}$$. It was demonstrated that all three expressions are equivalent, proving they are all correct solutions for the given integral.

Step-by-step solution

Write down the given integral and solution expressions

We are given that the following integral has three different solutions: $$\int \frac{dx}{x\sqrt{x^4 - 1}}$$ The expressions given by the computer algebra systems are: 1. \(\frac{1}{2} \cos^{-1} \sqrt{x^{-4}}\) 2. \(\frac{1}{2} \cos^{-1} x^{-2}\) 3. \(\frac{1}{2} \tan^{-1} \sqrt{x^4-1}\)

Simplify expressions

Now, let's simplify the first two expressions: 1. \(\frac{1}{2} \cos^{-1} \sqrt{x^{-4}} = \frac{1}{2} \cos^{-1} (x^{-2})\) 2. \(\frac{1}{2} \cos^{-1} x^{-2}\) In this case, the first expression is the same as the second expression after simplification.

Analyze the relationship between the remaining expressions

Now we need to understand the relationship between the remaining expressions: 1. \(\frac{1}{2} \cos^{-1} x^{-2}\) (Expression 1) 2. \(\frac{1}{2} \tan^{-1} \sqrt{x^4-1}\) (Expression 3) In order to show these are equivalent, we will use the properties of trigonometric functions such that: $$\cos^{-1} y = \tan^{-1}\sqrt{1 - y^2}$$ Now, plugging the "y" value from Expression 1 into the formula above: $$\frac{1}{2} \cos^{-1} x^{-2} = \frac{1}{2} \tan^{-1} \sqrt{1 - (x^{-2})^2}$$ $$\frac{1}{2} \cos^{-1} x^{-2} = \frac{1}{2} \tan^{-1} \sqrt{1 - x^{-4}}$$ Since \(x^4 - 1 = 1 - x^{-4}\), the expression becomes: $$\frac{1}{2} \cos^{-1} x^{-2} = \frac{1}{2} \tan^{-1} \sqrt{x^4 - 1}$$ Thus, Expression 1 and Expression 3 are equivalent, proving that all three expressions are correct solutions for the given integral.

Key concepts

Trigonometric Functions

Understanding trigonometric functions is crucial in integral calculus, especially when dealing with complex integrals that involve inverse trigonometric functions. These functions, like \(\cos^{-1}\) and \(\tan^{-1}\), are the inverse operations of their respective trigonometric functions, cosine and tangent.

Inverse trigonometric functions are used to solve equations where the angle is the unknown value. Instead of finding a side in a right triangle, they start with sides and find the measure of an angle. In the provided solution, we encounter \(\cos^{-1}\) and \(\tan^{-1}\), which represent angles whose cosine and tangent values, respectively, are given by certain expressions.

To better understand these inverse functions, remember:

• \(\cos^{-1}(x)\) finds the angle whose cosine is \(x\).
• \(\tan^{-1}(x)\) finds the angle whose tangent is \(x\).
• These functions are restricted to specific ranges: \(\cos^{-1}(x)\) generally between 0 and \(\pi\) while \(\tan^{-1}(x)\) ranges from \(-\pi/2\) to \(\pi/2\).

This restriction ensures that each angle maps to one unique value in the function's range, clarifying inverse operations.

Integration Techniques

Mastering integration techniques is key to addressing integrals that seem difficult at first. In the exercise, the integral \(\int \frac{dx}{x\sqrt{x^4 - 1}}\) can be intimidating, but with an understanding of techniques and relationships between expressions, the solution becomes accessible.

A crucial technique demonstrated here is recognizing equivalences in expressions using trigonometric identities. By understanding such identities, we can simplify complex expressions and see their equivalence. One such identity is \(\cos^{-1} y = \tan^{-1}\sqrt{1 - y^2}\), which shows a relationship between inverse cosine and inverse tangent.

To tackle integrals like in this exercise, follow these steps:

• Identify possible substitutions or trigonometric identities.
• Simplify expressions by recognizing patterns, such as \(1 - y^2 = x^4 - 1\). These transform complex integrals into simpler, solvable forms.
• Check the equivalence of final expressions to verify correct solutions.

By practicing these techniques, integrals involving complex expressions become more approachable and solvable.

Computer Algebra Systems

Computer Algebra Systems (CAS) are powerful tools that solve mathematical problems with precision. They use algorithms to compute complicated equations and typically provide solutions that can be examined and interpreted by users.

In the exercise, we discuss multiple solutions provided by CAS for an integral. Different systems can yield varied expressions for the same problem due to differing processes and simplifying techniques employed by each algorithm. Despite the differences, these solutions are often correct due to underlying mathematical equivalences, allowing different forms to represent the same result.

To make the most of CAS:

• Understand the basic principles of the functions and operations involved.
• Recognize that multiple correct answers can exist for complex problems—they may look different but represent similar mathematical truth.
• Always cross-reference CAS results with manual calculations when possible to ensure comprehension and verify solutions.

CAS showcases the beauty of math through the convergence of logic, allowing us to see multiple facets of a solution through different expressions.