Problem 45 - Essential Calculus

Question

In Exercises \(45-48,\) state whether you would use integration by parts to evaluate the integral. If so, identify what you would use for \(u\) and \(d v\). Explain your reasoning. $$ \int \frac{\ln x}{x} d x $$

Step-by-Step Solution

Answer

Yes, integration by parts can be used to evaluate the integral. \(u\) would be \( \ln x \) and \( dv \) would be \( \frac{1}{x} dx \), based on the ILATE rule and the derivative of \( \ln x \).

Step 1: Decide whether to use integration by parts

Here the integral is the product of \( \ln x \) and \( \frac{1}{x} \). The derivative of \( \ln x \) is \( \frac{1}{x} \) which can simplify the integral. Hence, it is a suitable problem to use integration by parts.

Step 2: Identify \(u\) and \(dv\)

In order to decide which function to choose for \(u\) and which one for \(dv\), use the ILATE rule (Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential). According to this rule, logarithmic function is chosen before an algebraic function. So, choose \(u = \ln x\) and \(dv = \frac{1}{x} dx\).

Step 3: Explain reasoning

The choice of \(u = \ln x\) and \(dv = \frac{1}{x} dx\) was made based on the ILATE rule and the fact that the derivative of \( \ln x \) is \( \frac{1}{x} \), which will make the resulting integral simpler when using the formula for integration by parts: \( \int u dv = u v - \int v du \)

Key Concepts

ILATE Rule

The ILATE rule is a mnemonic device that helps students remember the order of precedence when choosing functions for integration by parts, a technique used when an integral is a product of two functions that cannot be easily integrated as is. This rule suggests one should prioritize functions in the following order: Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, and Exponential functions.

When applying the ILATE rule, as seen in the example exercise, you identify the functions that are being multiplied. In the given integral, \( \ln x \) is a logarithmic function, and \( \frac{1}{x} \) can be considered algebraic. Using the ILATE rule, the logarithmic function takes precedence and is chosen as \(u\), while the algebraic function is selected as \(dv\). This strategic choice ensures that the resulting integral becomes simpler once the integration by parts formula is applied.

Integral Calculus

Integral calculus is a branch of mathematics that deals with the concept of an integral, which represents the accumulation of quantities and can be thought of as the area under a curve. The process of finding integrals is called integration. There are several techniques for performing integration, and integration by parts is one of them, used especially when the integrand is a product of two functions.

The fundamental idea behind integration by parts is to transform the integral of a product of two functions into a simpler form that can be more easily evaluated. The formula used is \( \int u dv = u v - \int v du \). A proper understanding of how to choose \( u \) and \( dv \) is crucial for simplifying and solving the integral efficiently. The successful application of this method can often hinge on recognizing patterns or relationships between functions that make categorization according to the ILATE rule beneficial.

Logarithmic Integration

Logarithmic integration involves integrating functions with logarithms. When the integral contains a logarithmic function, such as \( \ln x \), integration by parts is a useful strategy, as is the case with the given exercise. It is particularly helpful when the logarithmic function is multiplied by another function that is easier to differentiate, rather than to integrate.

By choosing the logarithmic function as \( u \), differentiating it simplifies the integrand because the derivative of \( \ln x \) is \( \frac{1}{x} \), and this is typically more straightforward to integrate. Hence, the complex task of integrating a logarithmic function becomes more manageable. Through the process of logarithmic integration, it is also common to use properties of logarithms, such as the product, quotient, and power rules, to manipulate the integrand into a more integrable form before applying the integration by parts technique.