Question

State whether you would use integration by parts to evaluate the integral. If so, identify u and dv. If not, describe the technique used to perform the integration without actually doing the problem. $$ \int x e^{x} d x $$

Short answer

Use integration by parts; choose \( u = x \) and \( dv = e^x \, dx \).

Step-by-step solution

Recognize the Integral Type - Polynomial and Exponential Product

The integral presented is \( \int x e^x \, dx \). It involves a product of a polynomial function \( x \) and an exponential function \( e^x \), which is a classic case for using integration by parts.

Recall the Integration by Parts Formula

Integration by parts is based on the formula: \[ \int u \, dv = uv - \int v \, du \]This method is ideal for integrals where one function is easily differentiable (e.g., a polynomial) and the other is easily integrable (e.g., an exponential).

Select Functions for u and dv

To use integration by parts, set \( u = x \) (because its derivative \( du = dx \) simplifies the process) and \( dv = e^x \, dx \) (since its integral \( v = e^x \) is straightforward).

Confirm Appropriateness of Integration Technique

With \( u = x \) and \( dv = e^x \, dx \), integration by parts is appropriate because the differentiation of \( u \) reduces the polynomial degree, simplifying the integral.

Key concepts

Polynomial Functions in Integration

Polynomial functions are expressions composed of variables and coefficients, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. A simple example of a polynomial function is \( x \), as seen in the integral \( \int x e^x \, dx \).

These functions are easy to differentiate— the derivative of \( x \) is just \( 1 \). This property makes polynomial functions uniquely suitable for certain integration techniques, especially integration by parts. When integrating a product that contains a polynomial function, choosing it as \( u \) in the integration by parts formula simplifies the process because its derivative \( du \) reduces its degree.

When working with polynomials in integrations, always estimate their behavior and degree. This will help make informed decisions on whether integration by parts or another method is more suitable for a given integral.

• Remember: When the polynomial is reduced to zero, the integration stops being necessary.
• For higher degree polynomials, multiple applications of integration by parts might be needed.

The Role of Exponential Functions

Exponential functions, such as \( e^x \), are powerful tools in calculus due to their unique properties. Unlike other functions, the derivative and the integral of the exponential function are both \( e^x \), which simplifies calculations and integrations tremendously.

In the context of integration, particularly integration by parts, exponential functions are often chosen as \( dv \) because its integral is direct and matches the original function. This makes it an ideal companion to polynomial functions within the product integral \( \int x e^x \, dx \).

Why is \( e^x \) so convenient here?

• It doesn't change form when differentiated or integrated.
• It doesn't introduce additional complex functions into the calculation.

This property makes exponential functions extremely useful, often appearing in integrations paired with other function types that are simplified by their presence.

Mastering Integration Techniques

Integration techniques offer a toolkit of methods and approaches to evaluate integrals efficiently. When you encounter an integral like \( \int x e^x \, dx \), it's important to recognize which approach will yield results more effectively.

The integration by parts technique is especially valuable for integrals involving products of different kinds of functions, like polynomials and exponentials. It can be thought of as the 'inverse' of the product rule for differentiation.

Key steps when using integration by parts:

• Select \( u \) and \( dv \) based on ease of differentiation and integration.
• Utilize the formula \( \int u \, dv = uv - \int v \, du \) to transform the integral.

Other methods, such as substitution, are beneficial when the integral can be directly simplified to a standard form. However, if two functions multiply each other with one being easily reduced and the other maintaining form, integration by parts is likely the best method.

Practicing different techniques on varied integrals can build a strong intuition for deciding which method suits best in different contexts.