Problem 36 - University Calculus: Early Transcendentals

Question

Evaluate the integrals. $$\int \frac{\cos \sqrt{\theta}}{\sqrt{\theta} \sin ^{2} \sqrt{\theta}} d \theta$$

Step-by-Step Solution

Answer

The integral evaluates to \( -2 \csc(\sqrt{\theta}) + C \).

Step 1: Substitute Variables

To simplify the integral, let's use a substitution. Set \( u = \sqrt{\theta} \). Then, we have \( \theta = u^2 \) and \( d\theta = 2u \, du \). This transforms our integral into \( \int \frac{\cos u}{u \sin^2 u} \cdot 2u \, du \).

Step 2: Simplify the Integral

After substitution, the integral becomes \( 2 \int \frac{\cos u}{\sin^2 u} \, du \). Notice that \( \frac{\cos u}{\sin^2 u} \) can be re-expressed as \( 2 \int \cot u \, \csc u \, du \).

Step 3: Integrate Using Known Identity

The integral \( \int \cot u \, \csc u \, du \) is a standard integral, and it equals \(-\csc u + C\) where \(C\) is a constant of integration. Thus, \( 2 \int \cot u \, \csc u \, du = -2 \csc u + C \).

Step 4: Substitute Back to Original Variables

Since we made the substitution \( u = \sqrt{\theta} \), substitute back to find the equation in terms of \( \theta \). Thus, \( -2 \csc u \) becomes \( -2 \csc(\sqrt{\theta}) \). Therefore, the integral evaluates to \( -2 \csc(\sqrt{\theta}) + C \).

Key Concepts

Substitution Method

The Substitution Method is often a powerful tool for solving integrals. It involves changing the variable in the integral to make the problem easier to solve. Here's how it works:

• Identify a part of the integrand that you can substitute with a single variable. In our example, we used the substitution \( u = \sqrt{\theta} \).
• Express \( \theta \) in terms of \( u \), which gives \( \theta = u^2 \).
• Find the derivative, which helps to replace \( d\theta \) as \( d\theta = 2u \, du \).

By substituting, we transformed a complicated integral into a simpler one with respect to \( u \), making it more manageable to solve.
Substitution can drastically simplify complex integrals if chosen wisely. It is important to substitute the correct element to simplify the problem without missteps.

Trigonometric Identities

Once substitution is complete, trigonometric identities often come into play, especially when dealing with trigonometric functions within integrals. In this exercise, we transformed our integrand to utilize properties of trigonometric identities:

• Recognize that \( \frac{\cos u}{\sin^2 u} \) can be rewritten using the identities \( \cot u = \frac{\cos u}{\sin u} \) and \( \csc u = \frac{1}{\sin u} \). Thus, our expression becomes \( \cot u \, \csc u \).

Utilizing identities like \( \sin^2 u + \cos^2 u = 1 \) or expressing functions in terms of \( \tan \), \( \cot \), \( \sec \), and \( \csc \) eases up calculations.
Mastering these identities is essential in order to manipulate trigonometric expressions into a form suitable for integration. This makes solving complex trigonometric integrals possible with confidence.

Standard Integrals

Standard integrals are those basic integrals whose solutions are widely recognized and used across different problems. They eliminate the need for extensive calculation once transformation through substitution or identities is achieved.

• In our problem, after substituting and simplifying, the integral \( \int \cot u \, \csc u \, du \) was identified as a standard integral.
• This standard integral has a known result: \( -\csc u + C \), where \( C \) represents the constant of integration.

Recognizing and applying standard integrals is crucial to effortlessly solving integrals without recalculating from scratch.
These standard integrals act as shortcuts, borrowing from solved problems of the past, and they serve as one of the best resources for quickly integrating recognized patterns and functions.