Question

In using the technique of integration by parts, you must carefully choose which expression is u. For each of the following problems, use the guidelines in this section to choose u. Do not evaluate the integrals. $$ \int x^{3} \ln (x) d x $$

Short answer

Choose \( u = \ln(x) \) and \( dv = x^3 \, dx \).

Step-by-step solution

Recognize the Integration By Parts Formula

The formula for integration by parts is given by: \[ \int u \, dv = u \, v - \int v \, du \] We must select parts of the integral so that one can be differentiated (\( u \)) and the other can be integrated (\( dv \)).

Identify the Parts of the Integrand

In the integral \( \int x^3 \ln(x) \, dx \), we have two components: \( x^3 \) and \( \ln(x) \). We need to decide which should be \( u \) and which should be \( dv \).

Use the LIATE Rule for Choosing u

The LIATE rule helps prioritize which function to choose as \( u \). It stands for Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential functions. \( \ln(x) \) is a logarithmic function, so it is prioritized over \( x^3 \), making \( u = \ln(x) \).

Assign u and dv

Based on the decision from the previous step, let \( u = \ln(x) \) which makes \( du = \frac{1}{x} \, dx \). Therefore, \( dv = x^3 \, dx \) and hence after integration, \( v = \frac{x^4}{4} \).

Setup Integration by Parts Formula

Substitute \( u = \ln(x) \), \( du = \frac{1}{x} \, dx \), \( v = \frac{x^4}{4} \), and \( dv = x^3 \, dx \) into the integration by parts formula: \[ \int x^3 \ln(x) \, dx = \ln(x) \cdot \frac{x^4}{4} - \int \left( \frac{x^4}{4} \cdot \frac{1}{x} \right) \, dx \] This prepares the integral for evaluation using the integration by parts method. However, evaluation is not required per the problem statement.

Key concepts

LIATE Rule

The LIATE Rule is a guideline for choosing the priority of functions when performing integration by parts. It's a helpful mnemonic that stands for:

• Logarithmic functions
• Inverse trigonometric functions
• Algebraic functions
• Trigonometric functions
• Exponential functions

The idea is to select the function that appears first on this list to be your \( u \). In the integral \( \int x^3 \ln(x) \, dx \), both \( x^3 \) (algebraic) and \( \ln(x) \) (logarithmic) are candidates. Logarithmic functions take precedence over algebraic ones, so \( \ln(x) \) is chosen as \( u \). This decision simplifies the process significantly because the chosen \( u \) will produce a manageable derivative for the integration process.

Integration Techniques

Integration techniques involve various methods to determine the integral of a function. In the context of integration by parts, this method extends from the product rule of differentiation.
The integration by parts formula is:

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

This formula shows us how to unravel an integral into simpler parts, provided we split our original integrand effectively into \( u \) and \( dv \).
For \( \int x^3 \ln(x) \, dx \), once \( u \) and \( dv \) are determined using the LIATE rule, the differentiation and integration required are straightforward calculations.

• Choose \( u = \ln(x) \).
• Then \( du = \frac{1}{x} \, dx \).
• Choose \( dv = x^3 \, dx \); thus \( v = \frac{x^4}{4} \).

By the method of integration by parts, integrals transform into forms where traditional or simpler methods can be easily applied.

Calculus Problem Solving

Calculus problem solving often requires choosing the right technique for each unique integral challenge. Effective problem solving in calculus involves several steps:

• Understanding the structure of the problem and choosing the right method, like integration by parts, substitution, or partial fractions.
• Identifying choices that simplify the problem. Using the LIATE rule for prioritization can make complex integrals manageable.
• Performing accurate calculations for differentiation and integration. These calculations demand precision, as errors can complicate rather than simplify the integrals.

For our integral \( \int x^3 \ln(x) \, dx \), strategizing with integration by parts illustrates this process. The LIATE rule guides the initial setup, leading to straightforward calculations and promoting an understanding of how calculus techniques work together for efficient problem-solving.