Question

For the following exercises, evaluate the integral using the specified method.\(\int \frac{\sqrt{4-\sin ^{2}(x)}}{\sin ^{2}(x)} \cos (x) d x\) using a table of integrals or a CAS

Short answer

Use a table of integrals or CAS to integrate and back substitute.

Step-by-step solution

Identify the Method

We are asked to evaluate the integral using a table of integrals or a Computer Algebra System (CAS), which means we need to identify a known integral formula or use software to find the solution.

Recognize Trigonometric Identity

Notice that the integrand contains the expression \( \sqrt{4 - \sin^2(x)} \). Using the Pythagorean identity, \( \cos^2(x) = 1 - \sin^2(x) \), we can express \( 4 - \sin^2(x) = 4\cos^2(x) + 3 \). This may help in simplifying the integral.

Substitution

To simplify, perform the substitution \( u = \sin(x) \). Then, \( du = \cos(x) \, dx \) and rewrite the integral in terms of \( u \): \( \int \frac{\sqrt{4-u^2}}{u^2} du \).

Simplify the Integral

Now, the integral can be simplified using the trigonometric identity \( \sqrt{4 - u^2} \to \sqrt{4}\sqrt{1 - (u/2)^2} = 2\sqrt{1 - (u/2)^2} \). The integral becomes \( 2 \int \frac{\sqrt{1-(u/2)^2}}{u^2} \, du \).

Consult Table of Integrals or CAS

Using a table of integrals or a CAS, find that \( \int \frac{\sqrt{a^2 - x^2}}{x^2} \, dx \) is a known form. For \( a = 2 \) and by matching our current form, use the table or software to integrate it.

Integrate and Back Substitute

After finding the integral in terms of \( u \), back substitute \( u = \sin(x) \) to express the solution in terms of \( x \). The evaluated integral using a CAS would yield the complete expression.

Key concepts

Trigonometric Substitution

Trigonometric substitution is a powerful technique in integral calculus that utilizes trigonometric identities to simplify integrals. By substituting trigonometric functions for variables, complex expressions can often be transformed into simpler ones. In our exercise, the trigonometric substitution is initiated with the relation \( u = \sin(x) \). This substitution is helpful because the Pythagorean identity \( \cos^2(x) = 1 - \sin^2(x) \) can simplify the integrand. The goal here is to reduce the integral expression so that it aligns with known integrals in a table or format used by a Computer Algebra System (CAS).
For example, when \( 4 - \sin^2(x) \) appears in the integrand, it can be rewritten using \( \cos^2(x) \) to facilitate the integration process. This technique breaks down complicated integrals, allowing us to manage them with tools like integral tables and CAS.

Pythagorean Identity

The Pythagorean identity is fundamental in trigonometry and essential in solving integrals involving trigonometric functions. The identity states that \( \sin^2(x) + \cos^2(x) = 1 \). This identity can be rearranged to express one trigonometric function in terms of another, such as \( \cos^2(x) = 1 - \sin^2(x) \).
In the exercise, this identity helps simplify the expression \( \sqrt{4 - \sin^2(x)} \) by converting it to \( \sqrt{4\cos^2(x) + 3} \). Although the appearance of terms might look more complex initially, it allows further simplification or substitution steps that align with known integral forms. Understanding and applying the Pythagorean identity enables us to transform and simplify integrals, a crucial skill in integral calculus.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is a software tool used to perform symbolic mathematics. It can handle tasks such as algebraic expressions manipulation, solving equations, and, notably, evaluating integrals. In situations where integrals are complex and difficult to tackle analytically, a CAS can assist in finding solutions efficiently.
Using a CAS is particularly beneficial when the integral aligns with a known form from integral tables, or when it needs a precise symbolic solution that involves numerous steps. In our exercise, after simplifying the integrand using trigonometric substitutions and identities, a CAS was used to find the integral's solution quickly. This approach combines advanced computing technology with mathematical techniques to deliver results that might be cumbersome to derive manually.

Integration Techniques

Integration techniques form the backbone of evaluating integrals in calculus. They include various methods like substitution, integration by parts, and trigonometric substitution, among others. Each technique serves to simplify the integration process depending on the integral's form.

• **Substitution**: Often used when an integral contains composite functions, enabling simplification by changing variables.
• **Trigonometric Substitution**: Useful for integrals involving square roots of quadratic expressions. It transforms complex expressions using trigonometric identities.
• **Tables of Integrals**: These provide standard forms of integrals with known solutions, allowing for quick referencing.

In our exercise, the integration started by using trigonometric substitution and ended with consulting a table or a CAS to determine the final integrated form. Mastering these techniques allows for evaluating a wide range of integrals by applying the most suitable method to the given problem.