Problem 24 - College Algebra and Calculus: An Applied Approach

Question

Find the indefinite integral. (Hint: Integration by parts is not required for all the integrals.) $$ \int x^{4} \ln x d x $$

Step-by-Step Solution

Answer

The indefinite integral of \(x^{4} \ln x\) is \(x^{5} \ln x /5 - x^{5}/25 + C\).

Step 1: Express the Integral in Different Parts

Express the function we want to integrate in two parts, \(u\) and \(v'\).\nFirst, we set \(u= \ln x\) and \(v' = x^{4}\). From this, we get that \(u' = 1/x\) and \(v = x^{5}/5\).

Step 2: Apply the Formula of Integration By Parts

We use the formula for integration by parts: \(\int u dv = uv - \int v du\). Substituting our functions and their derivatives in, we get \(\int x^{4} \ln x dx = x^{5} \ln x /5 - \int x^{4} dx /5\).

Step 3: Evaluate the Remaining Integral

The remaining integral, \(\int x^{4} dx\), must now be evaluated. It is equal to \(x^{5}/5\), so substituting this in yields \(\int x^{4} \ln x dx = x^{5} \ln x /5 - x^{5}/25 + C\), where C is the constant of integration.

Key Concepts

Integration by Parts

Integration by parts is a powerful method for evaluating integrals, especially when they involve products of functions. The principle is derived from the product rule of differentiation and involves choosing parts of the integrand to assign to different functions. In this method, we break down the function into two parts, typically denoted as\( u \) and\( dv \). The formula to remember is:

• \( \int u \, dv = u v - \int v \, du \)

First, you identify a function to be\( u \), which ideally simplifies upon differentiation, and another\( v' \), which should be easy to integrate.
In our example, for the integral \( \int x^{4} \ln x \, dx \), we recognize \( u = \ln x \) since its derivative is relatively simple and \( v' = x^4 \) as its integration is straightforward. By applying the formula, you transform a complex integral into simpler parts, finally reducing it to standard integrals for easy computation.

Logarithmic Functions

Logarithmic functions, such as the natural logarithm \( \ln x \), are a special class of functions used frequently in calculus and algebra. They exhibit unique properties that make them particularly useful in mathematical analysis. The natural log \( \ln x \) is the logarithm to the base \( e \), where \( e \) is an irrational number approximately equal to 2.71828.
Thy key property of logarithms includes their ability to transform products into sums: \( \ln(ab) = \ln a + \ln b \), and powers into products: \( \ln(a^b) = b\ln a \).
In integration, choosing \( u = \ln x \) is often effective because its derivative, \( u' = \frac{1}{x} \), is simple and simplifies the process.
This is why it's chosen within the integration by parts: it simplifies complexities associated with multiplying functions involving logs with polynomials.

Polynomial Functions

Polynomial functions are among the simplest yet most foundational functions in mathematics, characterized by terms like \( x^n \), where \( n \) is a non-negative integer. These functions are widely used due to their ease of differentiation and integration.

• The integral \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \) provides a straightforward approach to integrate polynomial terms, except in the case when \( n = -1 \).

This simplicity is evident in our integration problem, where \( v' = x^4 \). On integration, it becomes \( v = \frac{x^5}{5} \). This change from differentiation to integration is smooth, requiring you merely to increase the power by one and divide by the new power.
Understanding polynomial integration is crucial for evaluating integrals involving other complex expressions, as multiple polynomial functions serve as foundational pieces throughout various fields of mathematics and practical applications.