Problem 57 - Calculus Early Transcendentals: Pearson New International Edition

Question

derive the given reduction formula using integration by parts. $$ \int x^{\alpha} \cos \beta x d x=\frac{x^{\alpha} \sin \beta x}{\beta}-\frac{\alpha}{\beta} \int x^{\alpha-1} \sin \beta x d x $$

Step-by-Step Solution

Answer

Use integration by parts with \( u = x^\alpha \) and \( dv = \cos \beta x \, dx \) to derive the formula.

Step 1: Identify Functions for Integration by Parts

Integration by parts follows the formula \( \int u \, dv = uv - \int v \, du \). Choose \( u = x^{\alpha} \) and \( dv = \cos \beta x \, dx \). Then, differentiate and integrate to find \( du \) and \( v \).

Step 2: Differentiate and Integrate Chosen Functions

Differentiate \( u = x^{\alpha} \) to get \( du = \alpha x^{\alpha-1} \, dx \). Integrate \( dv = \cos \beta x \, dx \) to find \( v = \frac{\sin \beta x}{\beta} \).

Step 3: Apply Integration by Parts Formula

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

Step 4: Simplify the Equation

Simplify the expression: \( \int x^{\alpha} \cos \beta x \, dx = \frac{x^{\alpha} \sin \beta x}{\beta} - \frac{\alpha}{\beta} \int x^{\alpha-1} \sin \beta x \, dx \). This matches the reduction formula we aimed to derive.

Key Concepts

Reduction Formula

Reduction formulas are a powerful tool in solving complex integrals, especially when dealing with polynomials and trigonometric functions. They simplify the process by breaking down a complex integral into a sequence of simpler integrals. Each step in a reduction sequence reduces the power or degree of the variable, ultimately leading to an integral that is straightforward to solve. This is particularly useful in integration problems that involve higher powers or degrees, as seen in the integration of expressions like \( \int x^{\alpha} \cos \beta x \, dx \).The formula provides a systematic approach to solve these kinds of integrals by utilizing the integration by parts technique. Through the reduction formula derived in the example, we see how a complex integral is transformed into a simpler one, from \( \alpha \) to \( \alpha-1 \), reducing the power by 1 with each iteration. By repeatedly applying the formula, the integral eventually becomes simple enough to solve directly.

Differentiation

Differentiation is a fundamental concept in calculus. It involves calculating the derivative of a function, which provides the rate at which the function changes at any point. In the context of integration by parts, differentiation is a key step where we determine the derivative of our chosen function \( u \).Consider differentiating \( u = x^{\alpha} \). The differentiation with respect to \( x \) gives us \( du = \alpha x^{\alpha-1} \, dx \). This step is crucial because it allows us to compute the subsequent integral in the integration by parts formula.

• It breaks down the original function into simpler components.
• Facilitates the transition from a complex integral to a more manageable form.
• Enables the integral to be simplified iteratively.

Differentiation, therefore, is not just about finding the slope but about transforming the original problem into a version that's easier to handle.

Integration

Integration is the process of finding the integral of a function, essentially the reverse process of differentiation. It involves finding the area under a curve. In integration by parts, one of the functions you choose must be integrated, setting it apart from basic integration.In our example, after picking \( dv = \cos \beta x \, dx \), we integrate it to obtain \( v = \frac{\sin \beta x}{\beta} \). This integration step gives us a simpler expression to work with in the context of integration by parts.The integration in this context:

• Determines the antiderivative of a function component.
• Is essential for applying the integration by parts formula effectively.
• Transforms one part of the integral while the other part is being differentiated.

Understanding integration in this manner is pivotal, especially in complex integrals involving trigonometric or exponential functions.

Trigonometric Integration

Trigonometric integration is a specialized field of calculus that deals with integrating trigonometric functions. These integrals often appear in problems involving waves, oscillations, and other periodic phenomena. The example given highlights the integral of a polynomial times a cosine function, \( \int x^{\alpha} \cos \beta x \, dx \).The key to solving these types of integrals is using identities and techniques such as integration by parts, where trigonometric functions interact frequently. For instance, knowing that the integral of \( \cos \beta x \) results in \( \frac{\sin \beta x}{\beta} \) is a foundational step in reducing the complexity of the integral.Trigonometric identities and rules simplify the integration process:

• Convert between trigonometric functions using standard identities.
• Change integrals into forms that are straightforward to solve.
• Use reductions and substitutions to simplify repeated integrals.

This topic is essential for efficiently tackling problems where trigonometric functions are involved in calculus, particularly in engineering and physics applications.