Problem 51 - Thomas Calculus

Question

Use the substitution \(z=\tan (\theta / 2)\) to evaluate the integrals in Exercises 51 and \(52 .\) $$ \int \sec \theta d \theta $$

Step-by-Step Solution

Answer

The integral evaluates to \( \ln |\sec(\theta) + \tan(\theta)| + C.\)

Step 1: Substitution and Identity Transformations

Introduce the substitution \( z = \tan(\theta/2) \), then express \( \theta \) in terms of \( z \). The identities \( \tan(\theta) = \frac{2z}{1-z^2} \) and \( \sec(\theta) = \frac{1+z^2}{1-z^2} \) will be useful. Differentiate \( \theta \) to find \( d\theta \):\[ d\theta = \frac{2}{1+z^2} dz. \] Now rewrite \( \sec \theta \) in terms of \( z \).

Step 2: Transform the Integral

Substitute \( \sec(\theta) = \frac{1+z^2}{1-z^2} \) and \( d\theta = \frac{2}{1+z^2} dz \) into the integral: \[ \int \sec(\theta) d\theta = \int \frac{1+z^2}{1-z^2} \cdot \frac{2}{1+z^2} dz \] which simplifies to:\[ \int \frac{2}{1-z^2} dz.\]

Step 3: Partial Fraction Decomposition

The integral can be decomposed into partial fractions. The expression \( \frac{2}{1-z^2} \) can be written as:\[ \frac{2}{1-z^2} = \frac{1}{1-z} + \frac{1}{1+z}. \] Use this to split the integral into two simpler integrals:\[ \int \frac{1}{1-z} dz + \int \frac{1}{1+z} dz. \]

Step 4: Solve the Partial Fraction Integrals

Solve each integral separately:\[ \int \frac{1}{1-z} dz = -\ln|1-z| + C_1 \] and \[ \int \frac{1}{1+z} dz = \ln|1+z| + C_2. \] Combining the constants, the general solution is:\[ -\ln|1-z| + \ln|1+z| + C. \]

Step 5: Back-Substitute Terms in z to Theta

Substitute back \( z = \tan(\theta/2) \). The solution becomes:\[ \ln \left( \frac{|1+\tan(\theta/2)|}{|1-\tan(\theta/2)|} \right) + C. \] Use trigonometric identities to consolidate:\[ \ln |\sec(\theta) + \tan(\theta)| + C. \] This is the final form of the integral in terms of \( \theta \).

Key Concepts

Trigonometric Integrals

Trigonometric integrals are a fascinating area of calculus, focusing on the integration of trigonometric functions like sine, cosine, and tangent. When solving these, various techniques can simplify the process.
For example, using identities such as \(\sin^2(x) = 1 - \cos^2(x)\) can transform the function into a more manageable form, preparing it for integration.

• Integrating functions like \(\int \sec^n(x) dx\) might require more complex substitutions beyond common identities.
• These require manipulations or special substitutions, like interpreting the function as a rational expression often involving trigonometric identities.

Understanding these strategies allows you to navigate through trigonometric integrals more effectively, often requiring detailed rearrangements or complex substitutions.

Partial Fraction Decomposition

Partial fraction decomposition is a valuable tool to simplify integrals that involve rational expressions. When you encounter a fraction like \(\frac{2}{1-z^2}\), you can break it into simpler fractions that are easier to integrate.
Here's a step-by-step approach. First, express the rational function as a sum of fractions with linear numerators over linear or quadratic denominators when possible.

• For example, \(\frac{1}{1-z^2}\) can be decomposed into \(\frac{1}{1-z} + \frac{1}{1+z}\).
• This action separates the integral into two simpler parts.
• You then solve each part individually, leading to easier, straightforward operations.

This method is particularly useful when integrating, ensuring that the original complicated expression becomes simple and solvable through basic integration rules.

Substitution Method

The substitution method is a fundamental technique in calculus for simplifying the process of integration. By altering the variable of integration, this method often turns a complex problem into a simple one.
One of the most powerful substitutions in trigonometric integrals is using \(z = \tan(\theta/2)\). When you substitute, you also need to translate all expressions in terms of this new variable. Differentiating, you express \(d\theta\) in terms of \(dz\), ensuring a smooth transformation of the integral.

• Implement substitutions strategically to handle functions difficult to integrate directly.
• Always change back to the original variable after integration to maintain the context of the problem.

This approach is invaluable for easing complicated integrals, turning them into straightforward calculations. It also lays the groundwork for unraveling seemingly convoluted scenarios.

Secant Function Integration

The integration of the secant function, specifically \(\int \sec \theta d\theta\), is intriguing because it involves specific trigonometric transformations to arrive at a solution.
By using the substitution method with \(z = \tan(\theta/2)\), the integral can be transformed into a manageable algebraic form, specifically targeting strategies such as partial fraction decomposition.

• Initially, replace the secant function using known identities, like \(\sec(\theta) = \frac{1+z^2}{1-z^2}\).
• Coordinate the transformations to yield integrals ready for simple back-substitution.
• Transform back into terms of \(\theta\) by substituting \(\tan(\theta/2)\), ensuring you return to the initial function for the final result.

This approach provides insight into how auxiliary substitutions and transformations cooperate in the calculation of integrals involving secant, showing the elegance and complexity of calculus above basic trigonometric functions.