Problem 38 - Calculus: Early Transcendental Functions

Question

Evaluate the definite integral. $$\int_{1}^{\varepsilon} \frac{\ln x}{x} d x$$

Step-by-Step Solution

Answer

The value of the definite integral is \(1/2 * \ln^2(\varepsilon)\)

Step 1: Identify the variables for Integration by Parts

By using integration by parts, we identify \(u = \ln(x)\) and \(dv = 1/x\,dx\). Then by differentiating and integrating respectively, we find \(du = (1/x)dx\) and \(v = \ln(x)\).

Step 2: Apply the Integration by Parts Formula

The integration by parts formula is \(∫udv = uv - ∫vdu\). Substituting \(u, v, du\), and \(dv\), we obtain \(\int_{1}^{\varepsilon} \ln(x)/x\, dx = [\ln(x) * \ln(x)]_{1}^{\varepsilon} - \int_{1}^{\varepsilon} (\ln(x)*1/x)\, dx = [\ln^2(x)]_{1}^{\varepsilon} - \int_{1}^{\varepsilon} \ln(x)/x\, dx\). Now the integral on the right side is the same as the original integral, which means you can solve for it.

Step 3: Solve for the Integral

Setting the original integral equal to the result from the integration by parts gives \(2\int_{1}^{\varepsilon} \ln(x)/x\, dx = [\ln^2(x)]_{1}^{\varepsilon}\). Dividing both sides by 2 gives \(\int_{1}^{\varepsilon} \ln(x)/x\, dx = 1/2 * [\ln^2(\varepsilon) - \ln^2(1)]\). The natural log of 1 is 0, so this simplifies to \(1/2 * \ln^2(\varepsilon)\).

Key Concepts

Integration by Parts

Integration by parts is a powerful technique used in calculus to transform a complex integral into a more manageable form. This method is especially useful when integrating the product of two functions. It is derived from the product rule for differentiation. The basic formula for integration by parts is:

• \[ \int u \, dv = uv - \int v \, du \]

Here is how it generally works:

• Choose which part of the integrand is \(u\) and which is \(dv\).
• Differentiate \(u\) to get \(du\), and integrate \(dv\) to find \(v\).
• Substitute these into the integration by parts formula.
• This converts the original integral into another form that is often simpler to compute.

In the given exercise, we applied integration by parts by setting \(u = \ln(x)\) because the natural logarithm is often chosen for \(u\) as it simplifies upon differentiation. The differential element \(dv\) was then naturally \(1/x \, dx\), leading to simpler expressions for \(du\) and \(v\). This choice significantly simplified the problem because \(d(\ln(x)) = 1/x\, dx\) and \(v = \ln(x)\), allowing us to efficiently work through the step-by-step integration.

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is a fundamental concept in calculus. It serves as the inverse function of the exponential function \(e^x\). This makes it unique and invaluable, especially when dealing with exponential growth or decay.

• The natural logarithm has a base of \(e\), which is approximately 2.718.
• It helps in simplifying powers and products, allowing transformations like \(b^x = e^{x \, \ln(b)}\).
• One of its key properties: \(\ln(1) = 0\), which proves useful in solving integrals.

In the context of the problem, we chose \(u = \ln(x)\) in the integration by parts approach because it leads to easier differentiation into \(du = 1/x\, dx\). Also, evaluating boundaries for \(\ln(x)\) within the definite integral, such as\( \ln(1) = 0\), simplifies calculations, giving \(\ln^2(\varepsilon) - \ln^2(1)\), eventually simplifying to \(\ln^2(\varepsilon)\). This is a typical manipulation involving the natural logarithm, emphasizing its operational ease.

Calculus

Calculus is a branch of mathematics that deals with continuous change. It's divided mainly into differential and integral calculus. In this exercise, the focus is on integral calculus.

• Integral calculus is concerned with the accumulation of quantities, such as areas under curves, or more abstractly, integral equations.
• The definite integral \(\int_a^b f(x) \, dx\) represents the area under the curve \(f(x)\), from \(x = a\) to \(x = b\).
• Approaches like integration by parts allow us to handle more complex functions by breaking them into parts we know how to integrate.

The solution of the problem illustrates the application of calculus principles to compute the definite integral of \(\int_{1}^{\varepsilon} \frac{\ln x}{x} dx\). It involves selecting suitable techniques like integration by parts to reformulate the integral, leveraging properties of the natural logarithm, and interpreting the results within calculus' broader context. The integration techniques aim to determine exact values of areas or other properties represented mathematically, demonstrating the richness and power of understanding calculus.