Problem 61 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 55-61, derive the given reduction formula using integration by parts. $$ \begin{aligned} \int \cos ^{\alpha} \beta x d x &=\\\ & \frac{\cos ^{\alpha-1} \beta x \sin \beta x}{\alpha \beta}+\frac{\alpha-1}{\alpha} \int \cos ^{\alpha-2} \beta x d x \end{aligned} $$

Step-by-Step Solution

Answer

Derived using integration by parts, the reduction formula is: \[ \int \cos^{\alpha} \beta x \, dx = \frac{\cos^{\alpha-1} \beta x \sin \beta x}{\alpha \beta} + \frac{\alpha-1}{\alpha} \int \cos^{\alpha-2} \beta x \, dx \]."

Step 1: Identify the Components for Integration by Parts

The formula for integration by parts is \( \int u \, dv = uv - \int v \, du \). For the integral \( \int \cos^{\alpha} \beta x \, dx \), choose \( u = \cos^{\alpha-1} \beta x \) and \( dv = \cos \beta x \, dx \).

Step 2: Determine du and v

You've chosen \( u = \cos^{\alpha-1} \beta x \), so differentiate it: \( du = -(\alpha-1) \beta \cos^{\alpha-2} \beta x \sin \beta x \, dx \). For \( dv = \cos \beta x \, dx \), integrate it: \( v = \frac{\sin \beta x}{\beta} \).

Step 3: Apply Integration by Parts Formula

Substitute \( u \), \( v \), \( du \), and \( dv \) into the integration by parts formula: \[ \int \cos^{\alpha} \beta x \, dx = \cos^{\alpha-1} \beta x \cdot \frac{\sin \beta x}{\beta} - \int \frac{\sin \beta x}{\beta} \cdot -(\alpha-1) \beta \cos^{\alpha-2} \beta x \sin \beta x \, dx \].

Step 4: Simplify the Expression

Simplifying the expression, we have: \[ \frac{\cos^{\alpha-1} \beta x \sin \beta x}{\beta} - (\alpha-1) \int \cos^{\alpha-2} \beta x \sin^2 \beta x \, dx \].

Step 5: Express \( \sin^2 \beta x \) in Terms of \( \cos^2 \beta x \)

Use the identity \( \sin^2 \beta x = 1 - \cos^2 \beta x \) to rewrite the integral: \[ (\alpha-1) \int \left( \cos^{\alpha-2} \beta x - \cos^{\alpha} \beta x \right) \, dx \].

Step 6: Combine the Steps into the Required Formula

Substitute back to align with the reduction formula: \[ \int \cos^{\alpha} \beta x \, dx = \frac{\cos^{\alpha-1} \beta x \sin \beta x}{\alpha \beta} + \frac{\alpha-1}{\alpha} \int \cos^{\alpha-2} \beta x \, dx \].

Key Concepts

Integration by Parts

Integration by parts is a method used to solve integrals by reducing them into simpler parts. It comes from the product rule for differentiation, transforming the difficult process into a more manageable one. This technique is expressed using the formula:

• \( \int u \; dv = uv - \int v \; du \)

This formula allows us to split an integral involving a product of functions into simpler parts. In typical scenarios, one function (\( u \)) is chosen to be differentiated, while the other (\( dv \)) is integrated.
In the context of the original exercise, we chose \( u = \cos^{\alpha-1} \beta x \) and \( dv = \cos \beta x \; dx \). These selections simplify the integration process, helping us to derive a reduction formula efficiently. Differentiating \( u \) and integrating \( dv \), we can substitute into the integration by parts formula to find the integral's solution.
The core idea is to make the integral-solving process easier, transforming a complex problem into a simpler one and utilizing the power of algebraic manipulation.

Trigonometric Integrals

Trigonometric integrals involve functions like sine, cosine, and tangent, often raised to some power. These integrals can be tricky due to their cyclical nature and complex identities, but they can usually be simplified using trigonometric identities.
In our problem, we are working with an integral involving \( \cos^{\alpha} \beta x \). By using the identity \( \sin^2 \beta x = 1 - \cos^2 \beta x \), we can manipulate expressions to simplify the integration process. This identity helps transform the integral into a form where traditional techniques, like integration by parts, can be more effectively applied.
When dealing with trigonometric integrals, breaking these functions down using identities or rewriting them into an equivalent form can drastically reduce the complexity of the problem. Remember, trigonometric identities are your best friends here! They help in transforming the integral into a sum or difference of simpler integrals.

Derivation Techniques

Derivation techniques refer to the strategies used to derive formulas and solve equations. Especially in calculus, these techniques are essential in understanding how complex formulas come to be, like deriving the reduction formula in integrals.
In this particular context, the derivation of the reduction formula involved using integration by parts and trigonometric identities. By starting with a proper selection of \( u \) and \( dv \) for integration by parts, we turned the integral \( \int \cos^{\alpha} \beta x \; dx \) into a more recognizable form. Then, by applying algebraic and trigonometric manipulations, the reduction formula was formed.

• Selecting appropriate components for \( u \) and \( dv \)
• Differentiating and then simplifying using identities and algebra

These steps, while straightforward, rely heavily on a deep understanding of both algebraic manipulation and trigonometric properties. Students honing these derivation techniques will find themselves better equipped to tackle complex integrals and derive meaningful results.