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. $$ \int x^{3} e^{2 x} d x $$

Short answer

Choose \( u = x^3 \) and \( dv = e^{2x} \, dx \).

Step-by-step solution

Identify the Components for Integration by Parts

To use integration by parts, we start by identifying parts of the integral. Integration by parts is based on the formula \( \int u \, dv = uv - \int v \, du \). Our integral is \( \int x^3 e^{2x} \, dx \), so we need to choose \( u \) and \( dv \) appropriately.

Apply LIATE Rule for Choosing u

The LIATE rule helps in selecting \( u \), which suggests a preferred order: Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, and Exponential functions. In our integral, \( x^3 \) is an algebraic function and \( e^{2x} \) is an exponential function. According to the LIATE rule, algebraic functions come before exponential functions.

Choose u as the Algebraic Component

Following the LIATE rule, we choose \( u = x^3 \), which is the algebraic component, and thus fit our choice based on the guidelines for integration by parts.

Set dv as the Remaining Part

Since \( u = x^3 \), the remaining part of the integral \( dv \) is \( e^{2x} \, dx \). Thus, we have chosen \( dv = e^{2x} \, dx \). This will help in differentiating and integrating in the forthcoming steps once we solve fully.

Key concepts

LIATE Rule

One of the essential tools in the integration by parts method is the LIATE rule. This mnemonic helps us decide which component of a given integral should be chosen as the "u" in the formula \[ \int u \, dv = uv - \int v \, du \]. The letters in "LIATE" stand for various classes of functions, organized according to their diminishing priority:

• Logarithmic functions
• Inverse trigonometric functions
• Algebraic functions, such as polynomials
• Trigonometric functions
• Exponential functions

In any given integral problem, the components of the expression are compared against this list. The first component in the expression that appears higher on the LIATE list is typically chosen as "u."
In our exercise, we examined the integral \( \int x^3 e^{2x} \, dx \). Two functions stand out: \(x^3\), which is an algebraic function, and \(e^{2x}\), an exponential function. Based on the LIATE rule, we prioritize selecting \(x^3\) as "u" since it precedes exponential functions in priority.

Algebraic Functions

Algebraic functions play a central role in many mathematical operations, including integration. An algebraic function typically includes expressions that involve polynomial terms, such as \(x^3\) seen in our exercise.
When using integration by parts, algebraic functions are preferred candidates for selection as "u" because they simplify nicely when differentiated. The derivative of a polynomial function reduces the power of the terms, making the expression easier to handle in subsequent steps of solving an integral. For example, differentiating \(u = x^3\) with respect to \(x\) yields \(du = 3x^2 \, dx\).
In the context of our exercise, this reduction is crucial. While pairing with an exponential function like \(e^{2x}\), the algebraic nature of \(x^3\) ensures that when differentiated, the problem becomes more tractable rather than more complex.

Exponential Functions

Exponential functions are characterized by their form being \(e^{ax}\), where \(a\) is a constant. They are known for maintaining their form through differentiation and integration, as seen with \(e^{2x}\).
In integration by parts, exponential functions are often left to act as "dv" rather than being chosen as "u". This is because when you integrate an exponential function like \(e^{2x}\), it retains its basic structure, thereby simplifying the computations.
In our example \( \int x^3 e^{2x} \, dx \), while \(e^{2x}\) could be tempting to consider as "u" because of its simple manipulations, the algebraic function \(x^3\) is prioritized according to the LIATE rule. Thus, \(dv = e^{2x} \, dx\), ensuring that upon integration, the exponential function doesn't complicate the integration process but rather complements the differentiation of the algebraic function \(x^3\). This clever allocation aids in finding a neat and organized solution.