Problem 58 - Essential Calculus

Question

Use integration by parts to verify the formula. (For Exercises \(57-60\), assume that \(n\) is a positive integer.) $$ \int x^{n} \cos x d x=x^{n} \sin x-n \int x^{n-1} \sin x d x $$

Step-by-Step Solution

Answer

The integration by parts formula for the given problem is verified as \(\int x^n \cos x \, dx = x^n \sin x - n \int x^{n-1} \sin x \, dx\).

Step 1: Selecting Parts

Let the parts be chosen such that \(u = x^n\) and \(dv = \cos x \, dx\). For the integration by parts, the fundamental formula is \(\int udv = uv - \int vdu\). Here \(n\) is a constant so its differentiation would reduce the power by one, while the integral of \(\cos x\) is \(\sin x\).

Step 2: Differentiation and Integration

The next step involves differentiating and integrating the above chosen parts. Compute \(du\) by differentiating \(u=x^n\) and \(v\) by integrating \(dv=\cos x \, dx\). Thus, \(du = n \cdot x^{n-1} \, dx\) and \(v = \sin x\).

Step 3: Using Formula of Integration by Parts

Using the formula of integration by parts \(\int udv = uv - \int vdu\). Substituting \(u\), \(v\), \(du\) and \(dv\) we get, \(\int x^n \cos x \, dx = x^n \sin x - \int \sin x \, n \cdot x^{n-1} \, dx = x^n \sin x - n \int x^{n-1} \sin x \, dx\). The formula is verified.

Key Concepts

Integral Calculus

Integral calculus is one of the two main branches of calculus, alongside differential calculus, and focuses on the accumulation of quantities and the determination of the area under curves. This branch of mathematics primarily deals with integrals, which are essentially the reverse process of derivatives. Integrals help us find quantities such as areas, volumes, and other accumulative quantities.

There are two main types of integrals in integral calculus: definite and indefinite.

• Definite Integrals: These provide a numerical value representing the area under a curve between two points.
• Indefinite Integrals: These offer a general form solution, which includes a constant of integration, and represent a family of functions.

A core tool within integral calculus is the integration by parts method. This technique stems from the product rule of differentiation and is used to integrate products of functions. By breaking down functions into more manageable parts, integration by parts allows us to solve complex integrals that are not easily evaluated by standard means.

Definite and Indefinite Integrals

Definite and indefinite integrals are pivotal concepts in calculus that help us understand accumulation and the total value from a series of continuous values.

Definite Integrals: These integrals evaluate the exact area between a function and the x-axis within specified limits. They provide a numerical value, \([a, b]\), through \[ \int_{a}^{b} f(x) \, dx \]This helps calculate exact areas or total quantities in specific intervals.
Indefinite Integrals, on the other hand, focus on finding antiderivatives or the primitive form of a function. They do not have specified limits, leading to a general solution with a constant of integration. This constant symbolizes an unknown that can vary and makes indefinite integrals expressible as:
\[ \int f(x) \, dx = F(x) + C \]where \( C \) is the constant of integration. The term indefinite signifies the lack of boundaries, providing flexibility across a spectrum of solutions.

Mathematical Proofs

Mathematical proofs are logical arguments that establish the truth of mathematical statements. They are integral to mathematics, ensuring concepts are valid and universally accepted by demonstrating precision and rigor.

Mathematical proofs employ a series of logical steps to arrive at a conclusion. Proofs, such as those involved in verifying integration formulas, typically follow a straightforward process:

• Assumptions: Start with clear definitions and assumptions, such as stating properties of functions and behaviors like continuity.
• Logical Steps: Develop a series of logical deductions based on assumptions, previously established theorems, and axioms.
• Conclusion: Reach a final conclusion that summarizes what the proof demonstrates in clear terms.

In calculus, the application of proofs often involves verifying formulas or methods—like integration by parts—to ensure they provide accurate results. For instance, confirming a formula by substituting in values and showing each transformation adheres to the rules of calculus helps students internalize and trust mathematical processes. Proofs thus serve not only as a tool for verification but also as a method for deeply understanding mathematical logic and principles.