Problem 4 - College Algebra and Calculus: An Applied Approach

Question

Identify \(u\) and \(d v\) for finding the integral using integration by parts. (Do not evaluate the integral. $$ \int \ln 4 x d x $$

Step-by-Step Solution

Answer

The components \(u\) and \(d v\) for the given integral \( \int \ln 4 x d x \) are \(u = \ln 4x\) and \(d v = d x\).

Step 1: Identifying \(u\)

First, we identify the logarithmic part of the integrant, which should be chosen as \(u\). So, \(u = \ln 4x\).

Step 2: Identifying \(d v\)

Next, we identify the remaining part of the integrant which should be chosen as \(d v\). Here, the only part left is \(d x\). So, \(d v = d x\).

Key Concepts

Integral Calculus

Integral calculus is a fundamental part of calculus focusing on the concept of integration. Integration helps us find areas under curves, among other applications. Unlike differentiation, where we find the rate of change, integration involves finding the accumulation of quantities. It essentially "undoes" differentiation.

There are various techniques to perform integration, such as substitution and integration by parts, which simplify the integration process. Integral calculus is used in fields like physics, engineering, and economics to solve real-world problems.

• Definite Integrals: Represent the area under a curve between two limits and result in a specific number.
• Indefinite Integrals: Represent a family of functions and include a constant of integration, often denoted as '+ C'.

Understanding integral calculus forms the backbone of many advanced mathematics and applied science topics. It's essential for solving complex differential equations and modeling phenomena accurately.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. They are particularly important in mathematics due to their unique properties. A logarithmic function typically has the form \( \log_b(x) \), where \( b \) is the base. Natural logarithms, denoted as ln, specifically use the base \( e \) (approximately 2.718), a fundamental irrational number in mathematics.

Logarithmic functions are useful in simplifying equations where variables are in exponents. They help with solving growth and decay problems commonly found in science and economics.

• Properties of Logarithms: Include the product rule (\( \log_b(MN) = \log_b(M) + \log_b(N) \)), the quotient rule (\( \log_b(M/N) = \log_b(M) - \log_b(N) \)), and the power rule (\( \log_b(M^p) = p\log_b(M) \)).
• Change of Base Formula: Important for converting between different logarithmic bases: \( \log_b(x) = \frac{\log_k(x)}{\log_k(b)} \).

In the current exercise, the use of ln in the integrand points us towards selecting \( u \) as \( \ln 4x \), leveraging the properties of logarithms in integration by parts.

Calculus Techniques

Calculus techniques are essential tools for solving complex integrals and differentiation problems. Among them, one popular method for integration is the integration by parts technique.

Integration by parts is based on the product rule for differentiation. It is useful when the integration of a product of functions is involved and can be written as:\[\int u\, dv = uv - \int v\, du\]

For successful application, choosing the right \( u \) and \( dv \) is critical, as seen in the provided exercise. Here's how you identify them:

• Choose \( u \) as a function that becomes simpler when differentiated, like logarithms or polynomials.
• Select \( dv \) as the rest of the integrand.

Understanding when and how to apply integration by parts can simplify difficult integrals, transforming them into more manageable calculations. The exercise showed choosing \( u = \ln 4x \) and \( dv = dx \), highlighting the strategy to target the more complicated part with differentiation.