Problem 48 - Thomas Calculus

Question

Equations \((4)\) and \((5)\) lead to different formulas for the integral of \(\tan ^{-1} x :\) a. \(\int \tan ^{-1} x d x=x \tan ^{-1} x-\ln \sec \left(\tan ^{-1} x\right)+C\) b. \(\int \tan ^{-1} x d x=x \tan ^{-1} x-\ln \sqrt{1+x^{2}}+C\) Can both integrations be correct? Explain.

Step-by-Step Solution

Answer

Yes, both integrations are correct due to equivalent logarithmic identities.

Step 1: Identifying the Integral

We are given two different expressions for the integral of \( \tan^{-1} x \). Our task is to determine if both expressions can be correct. The integral form is generally written as \( \int \tan^{-1} x \, dx \).

Step 2: Check Function Equivalence Under Logarithmic Identities

First, we need to understand the different forms being presented. Notice that both alternatives involve natural logarithms: \( \ln \sec(\tan^{-1} x) \) and \( \ln \sqrt{1+x^2} \). Recognize that \( \sec(\tan^{-1} x) = \sqrt{1+x^2} \), which is derived from trigonometric and inverse trigonometric identities.

Step 3: Applying Trigonometric Identity

When \( y = \tan^{-1} x \), it implies \( x = \tan y \). Hence, \( \sec y = \sqrt{1 + \tan^2 y} = \sqrt{1 + x^2} \). Therefore, \( \ln \sec(y) = \ln \sqrt{1+x^2} \).

Step 4: Comparing Both Results

Since \( \ln \sec(\tan^{-1} x) \) is equivalent to \( \ln \sqrt{1+x^2} \), the expressions \( x \tan^{-1} x - \ln \sec(\tan^{-1} x) + C \) and \( x \tan^{-1} x - \ln \sqrt{1+x^2} + C \) are indeed equivalent, differing only in the representation of the logarithmic term.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools in calculus, particularly in integration problems. Unlike regular trigonometric functions, they take a ratio and return the angle. For example, \(\tan^{-1}x\) represents the angle whose tangent is \(x\). These functions are crucial because they help transform complex trigonometric relationships within integrals into simpler algebraic forms.

When solving integrals involving inverse trigonometric functions, we often use identities that relate them to algebraic expressions. These identities can simplify the problem and reveal insights about the integral's behavior.

• Understanding the range of inverse trig functions is important. For \(\tan^{-1}x\), the values fall between \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).
• Inverse trigonometric functions can be recognized through specific patterns in integrals, which can then be transformed using known algebraic identities.

Simplifying integrals with inverse trig functions often requires recognizing these patterns and applying the appropriate identity, as showcased in the original problem.

Logarithmic Identities

Logarithmic identities are important in manipulating and simplifying expressions in calculus, particularly within integral calculus. In the problem, two different expressions for a logarithmic term were analyzed: \(\ln \sec(\tan^{-1} x)\) and \(\ln \sqrt{1+x^2}\). This is a great example of how logarithmic identities can reveal equivalences.

One of the key tools is recognizing the identity \(\sec y = \sqrt{1 + x^2}\), derived from the trigonometric identity \(\sec^2 y = 1 + \tan^2 y\). By manipulating these expressions, we realize that they are equivalent, showing the inherent beauty and flexibility in calculus.

• Logarithmic identities like \(\ln(ab) = \ln a + \ln b\) and \(\ln(a^b) = b \ln a\) are fundamental in breaking down complex expressions.
• Understanding the proof methods for these identities can help solve integration problems more intuitively.

In the exercise solution, recognizing that \(\ln \sec(\tan^{-1} x) = \ln \sqrt{1+x^2}\) due to identity transformation further underscores their role in integrating inverse trigonometric functions.

Integration Techniques

Integration techniques extend beyond the basic rules and are fundamental in solving complex integrals involving inverse trigonometric functions and logarithms. It's about finding the right approach to handle seemingly complicated integrals.

In the given problem, recognizing the pattern and using an identity as a key technique led to resolving the integral equation. Techniques often involve substituting trigonometric identities or simplifying expressions as done through logarithmic identities.

The goal of these techniques is to simplify integrals into more readily solvable parts or standard forms that can be integrated directly.

• Substitution is a common technique where we replace a variable or expression with another to simplify the integral.
• Integration by parts is particularly useful with products of expressions, such as \(x \tan^{-1} x\) in the integral.
• Breaking down logarithmic and inverse trigonometric expressions can reveal easier paths to integration.

Utilizing integration techniques effectively requires practice in recognizing the setups, such as identifying counterparts in identities that simplify the integral expression.