Problem 41 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(13-48\) . $$ \int \frac{1}{\theta^{2}} \sin \frac{1}{\theta} \cos \frac{1}{\theta} d \theta $$

Step-by-Step Solution

Answer

Make a substitution and simplify; further simplification leads to a complex integral without an elementary antiderivative.

Step 1: Choose a Substitution

To simplify the integral \( \int \frac{1}{\theta^{2}} \sin \frac{1}{\theta} \cos \frac{1}{\theta} \, d\theta \), use the substitution \( u = \frac{1}{\theta} \). This implies that \( \theta = \frac{1}{u} \) and \( d\theta = -\frac{1}{u^2} \, du \).

Step 2: Substitute and Simplify

Substitute \( u = \frac{1}{\theta} \) and \( d\theta = -\frac{1}{u^2} \, du \) into the integral: \[ \int \sin u \cos u \left(-\frac{1}{u^2} \right) \, du = -\int \frac{\sin u \cos u}{u^2} \, du. \]

Step 3: Simplify Further Using Trigonometric Identities

Recognize \( \sin u \cos u = \frac{1}{2} \sin(2u) \) using the double-angle identity. This transforms the integral to:\[ -\int \frac{1}{2} \frac{\sin(2u)}{u^2} \, du = -\frac{1}{2}\int \frac{\sin(2u)}{u^2} \, du. \]

Step 4: Evaluate the Integral

This is a complex integral that typically requires power series expansion or numerical methods for a direct solution, as elementary antiderivatives are not easily applicable in this case. The integral \[ \int \frac{\sin(2u)}{u^2} \, du \]is a known form that can be addressed using more advanced calculus techniques or found in tables.

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a useful technique in integral calculus. It simplifies complex integrals involving square roots or trigonometric functions. It involves replacing a variable with a trigonometric expression. This transforms the integral into a more solvable form.
In the given exercise, a substitution technique is used where we set \( u = \frac{1}{\theta} \). This is instrumental because it turns the complex expression into one involving trigonometric terms. By substituting, we can employ trigonometric identities to further simplify the integral.

• Choose a substitution that makes the integral resemble a known form.
• Ensure that all parts of the original integral are expressed in terms of the new variable.

These substitutions help unlock further steps in solving, particularly when paired with algebraic manipulations.

Double-Angle Identity

The double-angle identity is a trigonometric identity used to rewrite expressions involving the sine and cosine of double angles. Specifically, the identity states that \( \sin(2u) = 2 \sin u \cos u \).
This identity is especially useful in the given exercise since it helps simplify the integral from an expression involving \( \sin u \cos u \) to one with just a single trigonometric function. This aids in reducing complexity and potentially making the integral easier to evaluate.

• Identify parts of the integral that match trigonometric identities.
• Apply the identity to simplify the terms in the integral.

Using such identities can be a critical step in handling difficult integrals.

Complex Integrals

Complex integrals refer to integrals that do not have straightforward solutions. They often involve intricate expressions or variable transformations.
In the example, the integral \[ \int \frac{\sin(2u)}{u^2} \, du \]is a complex one. It doesn't have a simple antiderivative in terms of basic functions, requiring advanced methods or numerical techniques.

• Recognize when an integral is complex and cannot be easily solved by hand.
• Explore options like power series expansion or lookup tables for solving.
• Consider employing numerical methods for approximation.

Complex integrals often demand creative solutions, underpinning the importance of understanding theoretical and practical tools in calculus.

Antiderivatives

Antiderivatives, or indefinite integrals, represent the reverse process of differentiation. They are fundamental in solving integrals that result in functions instead of constant values.
In the context of the exercise, finding an antiderivative for the complex integral can be challenging. The lack of an elementary antiderivative for \( \frac{\sin(2u)}{u^2} \) illustrates why certain integrals need special techniques.

• Understand that finding an antiderivative involves identifying a function whose derivative is the given expression.
• Realize not all integrals have elementary antiderivatives.
• Use advanced techniques or tools like integral tables when needed.

Recognizing when an antiderivative can be found, and when it requires advanced methods, is key in mastering integral calculus.