Problem 5 - Basic Analysis: Introduction to Real Analysis

Question

Prove integration by parts. That is, suppose \(F\) and \(G\) are continuously differentiable functions on \([a, b] .\) Then prove $$\int_{a}^{b} F(x) G^{\prime}(x) d x=F(b) G(b)-F(a) G(a)-\int_{a}^{b} F^{\prime}(x) G(x) d x.$$

Step-by-Step Solution

Answer

Integration by parts is derived from the product rule, rearranging gives the formula.

Step 1: State the Integration by Parts Formula

Integration by parts is a technique derived from the product rule for differentiation. It is expressed as:\[ \int F(x) G'(x) \, dx = F(x)G(x) - \int G(x)F'(x) \, dx \]}

Step 2: Recall the Product Rule for Differentiation

The product rule for differentiation states:\[ \frac{d}{dx} [F(x)G(x)] = F(x)G'(x) + G(x)F'(x) \].This will be the basis for deriving the integration by parts formula.

Step 3: Integrate the Product Rule over [a, b]

Integrate both sides of the product rule equation from \(a\) to \(b\):\[ \int_{a}^{b} \frac{d}{dx}[F(x)G(x)] \, dx = \int_a^b [F(x)G'(x) + G(x)F'(x)] \, dx \].

Step 4: Apply the Fundamental Theorem of Calculus

By the Fundamental Theorem of Calculus, the left side simplifies to the evaluation of \([F(x)G(x)]\) at the boundaries \(a\) and \(b\).\[ F(b)G(b) - F(a)G(a) = \int_a^b F(x)G'(x) \, dx + \int_a^b G(x)F'(x) \, dx \].

Step 5: Rearrange the Equation

Rearrange the equation to isolate the integral with \(F(x)G'(x)\):\[ \int_a^b F(x)G'(x) \, dx = F(b)G(b) - F(a)G(a) - \int_a^b G(x)F'(x) \, dx \].This is the integration by parts formula.

Key Concepts

Product Rule for Differentiation

When we think about the derivative of a product of two functions, we use the product rule. This is a fundamental concept in calculus. The product rule is formulated as follows: \[ \frac{d}{dx} [F(x)G(x)] = F(x)G'(x) + G(x)F'(x) \] Simply put, it tells us how to differentiate two functions that are multiplied together. Here's how it works:

• Consider two functions, \(F(x)\) and \(G(x)\).
• The derivative of their product is not simply the product of their derivatives.
• Instead, it involves each function and the derivative of the other function.

This rule is crucial for understanding more complex topics like integration by parts because it lays the groundwork for manipulating functions in calculus. By understanding and applying the product rule, we can explore different ways of expressing function products and their integrals.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, serving as a bridge between the two central operations in calculus. The theorem comes in two parts:

• The first part states that if \(F\) is an antiderivative of \(f\), then the integral of \(f(x)\) from \(a\) to \(b\) is \(F(b) - F(a)\).
• The second part asserts that if \(f\) is continuous on \([a, b]\), and \(F\) is defined by the integral of \(f\) from \(a\) to \(x\), then \(F'(x) = f(x)\).

In the context of integration by parts, the theorem is used to evaluate definite integrals. It tells us that the integral of the derivative of a function over an interval can be resolved into the difference between the values of the original function at the boundaries of the interval. Specifically, when applied to integration by parts, the theorem allows us to convert the integration of products into boundary evaluations and simpler integrals. This process helps simplify complex integrals and find solutions that might not be immediately apparent.

Continuously Differentiable Functions

Continuously differentiable functions are those that not only have derivatives, but do so smoothly and without interruption across their domain. In calculus, a function is said to be continuously differentiable over an interval if both the function and its derivative are continuous within that interval. Here are some key points:

• A function \(F(x)\) is continuously differentiable on \([a, b]\) if \(F'(x)\) exists and is continuous on \([a, b]\).
• It implies that there are no sudden jumps or breaks in the slope of the function.
• This smoothness is crucial for many calculus techniques, including integration by parts.

Continuous differentiability ensures that we can apply the theorems and techniques of calculus reliably. When functions are continuously differentiable, we can predict their behavior and confidently use operations like integration and differentiation without unexpected issues. In our integration by parts proof, the continuous differentiability of functions \(F\) and \(G\) guarantees that we can apply the product rule and the fundamental theorem effectively, leading to accurate and meaningful results.