Problem 1 - Calculus and its applications

Question

Find each antiderivative using Table 1. $$ \int x e^{-3 x} d x $$

Step-by-Step Solution

Answer

\( \int x e^{-3x} \, dx = -\frac{x + 1/3}{3} e^{-3x} + C \)

Step 1: Identify the Appropriate Integration Table Form

From Table 1, identify the integral form that resembles \( \int x e^{-3x} \, dx \). A matching form is \( \int x e^{ax} \, dx = \frac{1}{a^2}(ax - 1)e^{ax} + C \). In this instance, \( a = -3 \).

Step 2: Substitute the Known Values into the Formula

Using the formula from Step 1, substitute \( a = -3 \) into \( \int x e^{ax} \, dx = \frac{1}{a^2}(ax - 1)e^{ax} + C \), resulting in: \( \int x e^{-3x} \, dx = \frac{1}{(-3)^2}(-3x - 1)e^{-3x} + C \).

Step 3: Simplify the Expression

Calculate \( (-3)^2 \) and simplify the expression: \( \frac{1}{9}(-3x - 1)e^{-3x} + C \). Expand this to get \( \frac{-3x}{9} e^{-3x} - \frac{1}{9} e^{-3x} + C \).

Step 4: Final Simplicity Check

Simplify further if necessary: \( -\frac{x}{3} e^{-3x} - \frac{1}{9} e^{-3x} + C \), or finally \(-\frac{x + \frac{1}{3}}{3} e^{-3x} + C \).

Key Concepts

Understanding Integration Techniques

When you're tasked with finding an antiderivative, integration techniques are crucial to master. Knowing which technique to apply can simplify complex integrals considerably. In particular, recognizing a pattern within the integral is essential for selecting the correct integration technique.
Some popular techniques include:

• Substitution: Used when an integral contains a function and its derivative.
• Integration by Parts: Useful for products of functions, leveraging the formula: \[\int u \, dv = uv - \int v \, du.\]
• Partial Fractions: Ideal for integrals involving rational functions.

In this exercise, you employ integration by parts. It's tailored for integrals involving products, such as the one seen here: \[x e^{-3x}.\] The key is to spot that one part of the product simplifies into a form easier to integrate. With practice, recognizing these scenarios helps apply integration techniques effectively.

Integral Formulas: Your Mathematical Cheat Sheet

Integral formulas are like cheat sheets, guiding you through seemingly intricate problems. They provide straightforward pathways to solve integrals by recognizing specific patterns or forms. For this exercise, you rely on such a formula:
\[\int x e^{ax} \, dx = \frac{1}{a^2}(ax - 1)e^{ax} + C.\] This formula connects directly to the task of integrating \[x e^{-3x}. \] Recognizing this as a standard form, you directly substitute parameters to find the solution. These integral formulas save time and reduce errors in calculations by standardizing solutions for common integral types.
Integral tables consist of numerous such formulas, enabling rapid problem-solving by offering quick references to familiar patterns. Incorporating these formulas into your study can boost both confidence and efficiency in tackling calculus problems.

Exploring Exponential Functions in Integration

Exponential functions often appear in calculus exams due to their unique properties and are characterized by the base, usually \( e \), sitting at the heart of many key applications in mathematics and science. Understanding how to approach these functions is vital.
The function \[e^{ax} \] serves as the backbone for the integral you tackled. Unlike linear functions, the exponential function grows rapidly, complicating the integration process without proper formulas. This exercise demonstrates how integral formulas can simplify this process when encountering exponentials.
Here, by applying the specific integral formula: \[\int x e^{ax} \, dx, \] it guides you through the nuances of handling technologically and scientifically significant functions like exponentials. Mastering these allows you to appreciate their beauty and practicality, from solving differential equations to circuit analysis and more.