Question

Suppose \(g^{\prime}\) is integrable and \(f\) is continuous on \([a, b] .\) Show that \(\int_{a}^{b} f(x) d g(x)\) exists and equals \(\int_{a}^{b} f(x) g^{\prime}(x) d x\).

Short answer

Question: Show that if \(g'(x)\) is integrable and \(f(x)\) is continuous on the interval \([a, b]\), then the integral \(\int_{a}^{b} f(x) d g(x)\) exists and is equal to \(\int_{a}^{b} f(x) g'(x) dx\). Answer: The integral \(\int_{a}^{b} f(x) d g(x)\) exists and is equal to \(\int_{a}^{b} f(x) g'(x) dx\).

Step-by-step solution

Apply integration by parts formula

We will use the integration by parts formula, which is given by: $$ \int_{a}^{b} u(x) dv(x) = u(x)v(x)\Bigg|_a^b - \int_{a}^{b} v(x) du(x) $$ In our problem, let \(u(x) = f(x)\) and \(dv(x) = dg(x)\). To find \(du(x)\) and \(v(x)\), first differentiate and integrate, respectively: $$ du(x) = f'(x) dx $$ $$ v(x) = g(x) $$ Now, we can plug these into the integration by parts formula: $$\int_{a}^{b} f(x) d g(x) = f(x)g(x)\Bigg|_a^b - \int_{a}^{b} g(x) f'(x) dx $$

Differentiate \(g(x)\) to get \(g'(x)\)

Recall that \(g'(x)\) is the derivative of \(g(x)\). Differentiate \(g(x)\) to get: $$ g'(x) = \frac{d g(x)}{ dx } $$

Substitute \(dg(x)\) with \(g'(x) dx\) in the formula

Now, we can substitute \(dg(x)\) with \(g'(x) dx\) in the formula we derived in Step 1: $$ \int_{a}^{b} f(x) d g(x) = f(x)g(x)\Bigg|_a^b - \int_{a}^{b} g(x) f'(x) dx $$ Replace \(dg(x)\) with \(g'(x) dx\): $$ \int_{a}^{b} f(x) g'(x) dx = f(x)g(x)\Bigg|_a^b - \int_{a}^{b} g(x) f'(x) dx $$

Rearrange the equation to get the required result

We want to show that \(\int_{a}^{b} f(x) d g(x)\) exists and is equal to \(\int_{a}^{b} f(x) g'(x) dx\). So, let's isolate \(\int_{a}^{b} f(x) d g(x)\): $$ \int_{a}^{b} f(x) d g(x) = f(x)g(x)\Bigg|_a^b - \int_{a}^{b} g(x) f'(x) dx + \int_{a}^{b} f(x) g'(x) dx $$ Now, notice that \(\int_{a}^{b} g(x) f'(x) dx\) and \(\int_{a}^{b} f(x) g'(x) dx\) are both integrable because \(g(x)\) and \(f'(x)\) are integrable and continuous, respectively, on the interval \([a, b]\). Thus, their sum is also integrable: $$ \int_{a}^{b} f(x) d g(x) = f(x)g(x)\Bigg|_a^b + \int_{a}^{b} f(x) g'(x) dx - \int_{a}^{b} g(x) f'(x) dx $$ Therefore, we can conclude that \(\int_{a}^{b} f(x) d g(x)\) exists, and: $$ \int_{a}^{b} f(x) d g(x) = \int_{a}^{b} f(x) g'(x) dx $$

Key concepts

Real Analysis

Real analysis is a branch of mathematics that deals with real numbers and real-valued functions. This field encompasses the rigorous study of limits, continuity, integration, differentiation, and sequences and series. Real analysis provides the theoretical underpinning for calculus, which is the mathematical study of change and motion.

In the context of real analysis, integration, particularly the Riemann-Stieltjes integral, is a core concept. Integrals help us understand the accumulation of quantities, such as areas under curves or the total growth produced by a varying rate of change. Students learning real analysis will often start by understanding the Riemann integral – a fundamental method for quantifying the area beneath the graph of a function. The Riemann-Stieltjes integral generalizes this concept further, which brings us to our next key topic.

Riemann-Stieltjes Integral

The Riemann-Stieltjes integral is a generalization of the Riemann integral. Instead of integrating a function over an interval with respect to the length of the subintervals, which is the traditional approach, the Riemann-Stieltjes method integrates a function with respect to another function. This is particularly useful when dealing with distribution functions in probability theory or when the 'weight' of subintervals is determined by a different function than the one being integrated.

To compute a Riemann-Stieltjes integral, one sums up products of function values and increments of the second function (often denoted as the integrator) over partitions of the interval. As the partitions get finer, if the limit exists, one obtains the integral. The exercise provided involves integration by parts, which is a technique that can simplify the computation of Riemann-Stieltjes integrals, especially when dealing with products of functions, as we see with continuous functions.

Continuous Functions

A continuous function is one where small changes in the input produce small changes in the output; there are no sudden 'jumps' or 'holes' in the function's graph. A hallmark of continuous functions is that you can draw their graph without lifting your pen from the paper. Within real analysis, understanding continuous functions is crucial, as they have nice properties that make them easier to work with, especially in terms of limits, derivatives, and integrals.

The significance of continuity in our exercise is that it assures us the integral of a continuous function over a closed interval is well-defined. Furthermore, the integrand's continuity simplifies the problem, as it allows for the application of integration by parts without the concern of undefined behavior within the interval of integration. It's integral to note that the smooth behavior of these functions is a key aspect of why certain theorems and tools in real analysis, such as the fundamental theorem of calculus, apply.