Problem 19 - Thomas Calculus

Question

Use the table of integrals at the back of the book to evaluate the integrals. \(\int \frac{d \theta}{5+4 \sin 2 \theta}\)

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{2}{3} \tan^{-1}(3 \tan(\theta)) + C \).

Step 1: Recognize the Integral Form

First, observe the given integral, which is \( \int \frac{d \theta}{5 + 4 \sin 2 \theta} \). We recognize that this is a rational trigonometric integral, and we might find a direct match or use trigonometric identities to simplify it to a form that can be found in the table of integrals at the back of the book.

Step 2: Simplify Using Trigonometric Identities

To utilize the table of integrals effectively, recall the trigonometric identity for double angles: \( \sin 2\theta = 2 \sin \theta \cos \theta \). This might not directly simplify the denominator much further, but it's helpful for understanding the form we are dealing with, considering potential substitutions.

Step 3: Attempt Substitution

Considering the table might have a form matching \( A + B \sin x \), where a trigonometric substitution is used, consider setting \( u = 2\theta \), which implies \( du = 2 d\theta \) or \( d\theta = \frac{1}{2}du \). Rewriting the integral in terms of \( u \), we have \( \int \frac{1}{2} \frac{du}{5 + 4 \sin u} \).

Step 4: Match the Integral to a Known Form

Look for a matching form in the table of integrals. Typically, tables include forms of \( \int \frac{du}{a + b \sin u} \), similar to \( \int \frac{du}{A + B \sin u} \) which can typically be integrated using a specific formula found in these tables.

Step 5: Apply the Formula

According to the integral table, \( \int \frac{du}{a + b \sin u} = \frac{2}{\sqrt{a^2 - b^2}} \cdot \text{tan}^{-1}\left(\frac{\sqrt{a^2 - b^2} \cdot \tan\left(\frac{u}{2}\right)}{a - b}\right) + C \), where \( a = 5 \) and \( b = 4 \).Since \( a^2 - b^2 = 5^2 - 4^2 = 25 - 16 = 9 \), the expression simplifies to:\[ \frac{2}{3} \cdot \text{tan}^{-1}\left(\frac{3 \cdot \tan\left(\frac{u}{2}\right)}{5 - 4}\right) + C \]

Step 6: Substitute Back to Original Variable

Substitute back \( u = 2\theta \) into the resulting expression to get the integral in terms of \( \theta \):\[ \int \frac{d\theta}{5 + 4 \sin 2 \theta} = \frac{2}{3} \cdot \text{tan}^{-1}\left(\frac{3 \cdot \tan\left(\theta\right)}{1}\right) + C \]

Key Concepts

Integration Techniques

When tackling integrals involving trigonometric functions, it's often necessary to leverage various integration techniques for simplification and evaluation. Common methods include:

• Direct Integration: When the integral's format directly matches a form in standard tables or known results, allowing for immediate evaluation.


• Algebraic Simplification: Using algebraic identities to alter the expression within the integral into a more manageable form.


• Trigonometric Identities: Applying identities to transform the integral, for instance, double angle identities like \( \sin 2\theta = 2 \sin \theta \cos \theta \), which can adjust the integral's form to fit within known results.

These techniques streamline solving integrals by either directly matching them to known forms or transforming them to do so. The goal is to make the integral easier to handle either by simplification or by substitution, preparing it for completion using tables or known results.

Trigonometric Substitution

Trigonometric substitution is a powerful tool when dealing with integrals involving trigonometric functions. By changing the variable of integration to a trigonometric function, the integration process often becomes simpler. For example, consider an integral where the substitution of a variable could simplify the trigonometric expressions involved, such as:

• Using \( u = 2\theta \), transforming \( \sin 2\theta\) to \( 2 \sin \theta \cos \theta \).


• The derivative \( du = 2 d\theta \) aids in switching the variable from \( \theta \) to \( u \).


• The transformed integral in terms of \( u \) may more easily match known forms in a table of integrals.

Applying the correct substitution not only adjusts the limits or format of the function inside the integral but also makes it conform to standard forms found in integral tables, simplifying the computation process immensely.

Table of Integrals

Tables of integrals are invaluable resources for quickly solving complexity within integration problems. These tables contain a wide array of integral forms with their corresponding evaluated solutions. When using a table of integrals, ensure:

• Form Identification: Look for patterns or recognizable forms within your integral that match entries in the table, such as \( \frac{1}{a + b \sin x} \).


• Parameter Mapping: Identify and map constants from your integral to the form found in the table, as demonstrated with \( a = 5 \) and \( b = 4 \) where applicable.


• Formula Application: Apply the relevant formula from the table. For example, using \( \int \frac{du}{a + b \sin u} \) results in an arctan form involving \( a \) and \( b \).

Using a table effectively requires familiarity with standard forms and some trigonometric simplification skills. This significantly reduces computational time and aids in confirming the correctness of indefinite integrals combined to obtain a verified solution.