Problem 20 - Calculus Single Variable

Question

Use the method of substitution to calculate the indefinite integrals. $$ \int \frac{24 \sin (x)}{\sqrt{2+\cos (x)}} d x $$

Step-by-Step Solution

Answer

The integral is \(-48 \sqrt{2 + \cos(x)} + C\).

Step 1: Choose a Substitution

Since the integrand has \( \sqrt{2 + \cos(x)} \), a good substitution would aim to simplify this expression. Let \( u = 2 + \cos(x) \). This substitution will simplify the square root.

Step 2: Find the Derivative of Substitution

Differentiate the substitution with respect to \( x \). The derivative is \( \frac{du}{dx} = -\sin(x) \). Rearrange to express \( dx \): \( dx = -\frac{du}{\sin(x)} \).

Step 3: Substitute in the Integral

Substitute \( u \) and \( dx \) into the integral. We have:\[ \int \frac{24 \sin(x)}{\sqrt{u}} \left(-\frac{du}{\sin(x)}\right) = \int \frac{-24}{\sqrt{u}} \, du \].

Step 4: Simplify the Integral

The integral simplifies to:\[ -24 \int \frac{1}{\sqrt{u}} \, du \].This simplifies to \(-24 \int u^{-1/2} \, du\).

Step 5: Integrate with Respect to \( u \)

Integrate \( u^{-1/2} \) with respect to \( u \):\[ -24 \int u^{-1/2} \, du = -24 \left( 2u^{1/2} \right) + C \ = -48u^{1/2} + C \].

Step 6: Substitute Back \( u = 2 + \cos(x) \)

Return to the variable \( x \) by substituting back \( u = 2 + \cos(x) \):\[ -48 \sqrt{2 + \cos(x)} + C \].

Key Concepts

Substitution Method

The substitution method is a powerful technique used in calculus for evaluating integrals. It allows you to simplify the integral by changing the variable, often transforming a complex expression into an easier one to integrate.
In the given exercise, the substitution method is used to handle the complex expression \( \frac{24 \sin(x)}{\sqrt{2+\cos(x)}} \). By introducing a new variable \( u = 2 + \cos(x) \), we effectively reduce the complexity of the square root, making the integral more straightforward to solve.
Key points to remember when using the substitution method:

• Choose a substitution that simplifies the integral.
• Differentiating your substitution helps in rewriting the differential \( dx \) in terms of \( du \).
• Always substitute back the original variable at the end of the integration process.

This technique is particularly useful when the integrand contains a composite function, or where a substitution transforms the integral into a basic form that is easy to evaluate.

Trigonometric Integration

Trigonometric integration involves integrating functions containing trigonometric functions such as \( \sin(x) \), \( \cos(x) \), \( \tan(x) \), and others. These types of integrals are frequent in calculus, especially when dealing with periodic functions.
In this particular example, the function \( \frac{24 \sin(x)}{\sqrt{2+\cos(x)}} \) involves both sine and cosine, common trigonometric functions. The goal here is to carefully select substitutions or transformations that help simplify the trigonometric expressions.
A few strategies for tackling trigonometric integration include:

• Using trigonometric identities to simplify expressions.
• Applying substitution to simplify the integration of trigonometric functions.
• Understanding the derivatives of trigonometric functions, as they guide the choice of substitution.

Mastering these strategies improves the ability to handle a wide variety of problems involving trigonometric integrals.

U-Substitution

U-substitution is a specific form of the substitution method widely used when integrating. It is particularly effective when dealing with compositions of functions where direct integration is challenging.
In the exercise given, \( u = 2 + \cos(x) \) serves as the substitution. This choice helps simplify the integral by allowing us to express the square root and other complex functions in terms of \( u \) instead of \( x \).
Steps to apply u-substitution effectively involve:

• Identifying a part of the integrand that can be substituted with \( u \).
• Deriving \( du \) and restructuring the differential \( dx \) to fit \( du \).
• Replacing all \( x \)-dependent parts of the integral with \( u \).

U-substitution transforms the integration process into something more manageable, especially for functions that involve inner functions or compositions.

Calculus Problem Solving

Solving calculus problems involving integration requires a strategic approach and understanding of different techniques. Indefinite integrals call for an antiderivative and often a constant of integration, \( C \).
Given the problem \( \int \frac{24 \sin(x)}{\sqrt{2+\cos(x)}} dx \), breaking it down using the substitution method and integrating in terms of a new variable is key to solving it effectively.
Important steps in calculus problem solving include:

• Analyzing the integrand and deciding on the best method of integration.
• Making suitable substitutions to simplify the integral.
• Carefully integrating the new function in terms of the new variable.
• Re-substituting the original variable and simplifying where possible.

The process requires careful manipulation and understanding of integral properties to arrive at a solution that is precise and comprehensible.