Problem 61 - Calculus Single Variable

Question

Calculate the integrals. $$ \int \frac{\cos (x)}{\sqrt{4-\sin ^{2}(x)}} d x $$

Step-by-Step Solution

Answer

The integral simplifies to \( 2\sin(u) + C \) using appropriate substitutions.

Step 1: Identify Substitution

Notice that the integral contains an expression of the form \( \sqrt{4 - \sin^2(x)} \). This resembles the structure \( \sqrt{a^2 - x^2} \). Recognizing a trigonometric identity can help; \( \cos^2(x) + \sin^2(x) = 1 \) can be re-arranged as \( \cos^2(x) = 1 - \sin^2(x) \). Therefore, consider using a trigonometric substitution to simplify the integral.

Step 2: Apply Trigonometric Substitution

To match the structure \( \sqrt{a^2 - x^2} \), notice that \( 4 - \sin^2(x) = 4\cos^2(x) \). Therefore, substitute \( \sin(x) = 2\sin(u) \) which implies \( \cos(x) = 2\cos(u) \), and change the limits accordingly where \( dx = 2\cos(u) du \). The integral becomes \( \int \frac{2\cos(u) \cdot 2\cos(u) du}{2\cos(u)} \), simplifying to \( 2\int \cos(u) du \).

Step 3: Simplify and Integrate

We now have the simplified integral \( 2\int \cos(u) du \). The antiderivative of \( \cos(u) \) is \( \sin(u) \). Integrating, we get \( 2\sin(u) + C \).

Step 4: Revert Substitution

Substitute back using the original substitution \( \sin(x) = 2\sin(u) \) or \( \sin(u) = \frac{1}{2}\sin(x) \). Thus, we have \( \sin(u) = \cos^{-1}(\frac{1}{2}\sin(x)) \). Substitute \( u = \sin^{-1}(\frac{\sin(x)}{2}) \) back into the expression, resulting in \( 2 \sin(u) = 2 \sin(\sin^{-1}(\frac{\sin(x)}{2})) = \sin(x) \), giving us the antiderivative.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools used in mathematics to simplify expressions and solve equations.
In this exercise, the identity \( \cos^2(x) + \sin^2(x) = 1 \) plays a crucial role.**Why Use Trigonometric Identities?**

• They help transform complex trigonometric expressions into simpler ones.
• They facilitate the process of integration or differentiation by reducing complicated terms.

When you encounter an integral involving expressions like \( \sqrt{4 - \sin^2(x)} \), recognizing the pattern similar to \( \sqrt{a^2 - x^2} \) is key.**Key Concepts**

• Replacing \( 1 - \sin^2(x) \) with \( \cos^2(x) \) simplifies your expression significantly.
• Knowing different trigonometric identities allows more versatility in manipulating equations.

These identities are especially valuable when performing substitutions in calculus, as demonstrated in this problem.

Integration Techniques

Integration techniques are methods used to find the integral of functions that cannot be directly integrated using basic formulas.
The exercise employs a specific technique known as trigonometric substitution, which involves substituting one trigonometric function for another to simplify an integral.**Trigonometric Substitution**

• This technique is particularly useful for integrals involving square roots of the form \( \sqrt{a^2 - x^2} \), \( \sqrt{a^2 + x^2} \), or \( \sqrt{x^2 - a^2} \).
• In this exercise, recognizing \( 4 - \sin^2(x) \) can be rewritten using trigonometric identities allows us to substitute \( \sin(x) = 2\sin(u) \).

The goal is to transform the original integral into a simpler form that is easily solvable, such as substituting \( x \) with an expression involving \( u \), resulting in a manageable function to integrate.**Steps in Trigonometric Substitution**1. Identify the form of the expression and choose the appropriate trigonometric identity.2. Make the substitution and express everything in terms of the new variable \( u \).3. Simplify the integral and solve.By following these steps, the integral becomes a straightforward process, as shown in this solution.

Antiderivatives

An antiderivative, also known as an indefinite integral, is the reverse process of differentiation.
Finding the antiderivative means determining the original function from its derivative.**Understanding Antiderivatives**

• The antiderivative of \( \cos(x) \) is \( \sin(x) + C \), where \( C \) is the constant of integration.
• When integrals are solved, the addition of \( C \) accounts for all possible vertical shifts of the function.

In this problem, after substituting and simplifying, the focus shifts to integrating \( \cos(u) \).**Steps to Solve for an Antiderivative**1. After substitution and simplification, recognize the integrand, such as \( \cos(u) \).2. Integrate the function to find the antiderivative in terms of \( u \).3. Revert back to the original variable, if a substitution was made.4. Add the constant \( C \) to denote the general solution.This problem illustrates how substitution and recognition of \( \cos \) forms lead directly to determining the antiderivative, completing the solution elegantly.