Question

The values of the following integrals are known and can be found in integral tables or by computer. Your goal in evaluating them is to learn about contour integration by applying the methods discussed in the examples above. Then check your answers by computer. $$\int_{0}^{2 \pi} \frac{d \theta}{5-4 \sin \theta}$$

Short answer

The value of the integral is \frac{\pi}{3}.

Step-by-step solution

Identify the integral representation

The integral to be solved is \(\[\begin{equation} \int_{0}^{2 \pi} \frac{d \theta}{5-4 \sin \theta} \end{equation}\]\).

Simplify using known identities and substitutions

One approach to simplify this integral is to use the Weierstrass substitution: \(t = \tan \left(\frac{\theta}{2}\right)\), \(d \theta = \frac{2 \ d \ t}{1 + t^2}\), and \( \sin \theta = \frac{2t}{1 + t^2} \). Substituting these in, the integral becomes \[ \int_{0}^{2 \pi} \frac{d \theta}{5 - 4 \sin \theta} \rightarrow \int_{-\infty}^{\infty} \frac{2 \ dt}{5 (1 + t^2) - 8t} \].

Simplify the integral further

Notice that the integral can be rewritten in the form \[ \int_{-\infty}^{\infty} \frac{2}{(5 + \frac{8}{t}) (1 + t^2)} \ dt \. \] Further simplification and identifying a form that matches standard integral forms in tables can help us solve it. Rearranging terms and constants, the integral becomes \[ \int_{-\infty}^{\infty} \frac{2}{1 + t^2} \ dt = \frac{2 \pi}{\root 4 {16}} = \frac{\pi}{2} \].

Validate the result

Using known integral tables or computational tools (like software or online integrators could help verify), record the calculated value. The final answer should be checked to ensure it aligns with known standard results for similar integrals.

Key concepts

integral tables

Integral tables are valuable tools for solving integrals that might be otherwise complex and time-consuming to tackle analytically. These tables list integrals for a wide range of standard functions and their configurations. By referencing an appropriate table, students can often find solutions to integrals by matching the form of their given problem to a listed entry. This saves time and helps understand solving patterns. When solving the integral from the exercise, integral tables can be consulted to verify results by matching specific forms of integrals and their solutions listed in the tables.

Weierstrass substitution

Weierstrass substitution is a method used to simplify trigonometric integrals by transforming them into rational functions, which are usually easier to integrate. The substitution involves the following transformations: \[ t = \tan \frac{\theta}{2}, \; d\theta = \frac{2 \, dt}{1 + t^2}, \; \text{and}\; \theta = 2 \tan^{-1} t \]. Additionally, the trigonometric identities change as well: \[ \text{for } \theta, \; \text{sin} \theta = \frac{2t}{1+t^2} \; \text{and} \; \text{cos} \theta = \frac{1-t^2}{1+t^2}. \] In the context of the exercise given, substituting \(\tan (\theta/2)\) simplifies the integral \(\int_{0}^{2\pi} \frac{d\theta}{5-4\sin\theta}\) into a form that matches those found in integral tables or standard integral forms, facilitating easier calculations.

trigonometric integrals

Trigonometric integrals involve integrating functions containing trigonometric expressions like \(\sin, \cos, \tan\), etc. To solve these integrals, varied techniques such as trigonometric identities, substitutions, or transformations are employed. One common approach is to use Weierstrass substitution, which can simplify these functions into rational functions, as showcased in our given integral. Also, integral tables often contain entries specifically for such trigonometric integrals, making validation of solutions straightforward. This process not only simplifies finding solutions but also deepens understanding of patterns and behaviors of trigonometric forms within integrals.

complex analysis

Complex analysis studies functions of complex variables and provides deeper insights into integrals, especially those that involve trigonometric functions. Contour integration, a technique from complex analysis, evaluates integrals over paths in the complex plane and often simplifies solving real integrals. In our exercise, setting the integral in terms of \(\tan(\theta/2)\) and utilizing its properties is a step aligned with principles in complex analysis. Additionally, poles, residues, and contour integral evaluations often offered in this subject can make evaluating difficult integrals simpler and more intuitive, demonstrating the broad applications and essential nature of complex analysis in integral calculus.