Problem 84 - Applied Calculus

Question

Repeated Integration by Parts Using a Table The solution to a repeated integration by parts problem can be organized in a table. As an example, we solve \(\int x^{2} e^{3 x} d x .\) We begin by choosing $$ u=x^{2} \quad d v=v^{\prime} d x=e^{3 \tau} d x $$ We then make a table consisting of the following three columns: Finally, the solution is found by adding the signed products of the diagonals shown in the table: $$ \int x^{2} e^{3 x} d x=\frac{1}{3} x^{2} e^{3 x}-\frac{2}{9} x e^{3 x}+\frac{2}{27} e^{3 x}+C $$ After reading the preceding explanation, find each integral by repeated integration by parts using a table. \(\int x^{2} e^{2 x} d x\)

Step-by-Step Solution

Answer

\( \frac{1}{2} x^2 e^{2x} - \frac{1}{2} x e^{2x} + \frac{1}{2} e^{2x} + C \)

Step 1: Select Functions for Integration by Parts

For repeated integration by parts, we start by selecting \( u \) and \( dv \). Let's choose \( u = x^2 \) and \( dv = e^{2x} \, dx \). By doing this, we can easily reduce the power of \( x \) in each iteration.

Step 2: Differentiate and Integrate the Parts

Differentiate \( u \): \[ du = 2x \, dx \] Integrate \( dv \):\[ v = \frac{1}{2} e^{2x} \] We'll fill these values into our table later.

Step 3: Create the Integration by Parts Table

Create a table with three columns: Function \( u \), Derivative \( du \), and Integral \( v \). Fill in the row using the functions:1. \( u = x^2 \), \( du = 2x \), \( v = \frac{1}{2} e^{2x} \)2. \( u = 2x \), \( du = 2 \), \( v = \frac{1}{4} e^{2x} \)3. \( u = 2 \), \( du = 0 \), \( v = \frac{1}{4} e^{2x} \)

Step 4: Calculate the Products of the Diagonals

Use the table to find the products of the diagonals and add them together, remembering to alternate signs:- First diagonal: \( u \cdot v = x^2 \cdot \frac{1}{2} e^{2x} = \frac{1}{2} x^2 e^{2x} \)- Second diagonal: \( u \cdot v = 2x \cdot \frac{1}{4} e^{2x} = \frac{1}{2} x e^{2x} \) (negative: \(-\frac{1}{2} x e^{2x}\))- Third diagonal: \( u \cdot v = 2 \cdot \frac{1}{4} e^{2x} = \frac{1}{2} e^{2x} \) (positive: \(\frac{1}{2} e^{2x}\))

Step 5: Combine the Results

Sum the signed products of the diagonals to get the final integral result:\[ \int x^2 e^{2x} \, dx = \frac{1}{2} x^2 e^{2x} - \frac{1}{2} x e^{2x} + \frac{1}{2} e^{2x} + C \]

Key Concepts

Integral Calculus

Integral calculus is a branch of mathematics that deals with the concept of accumulation. It focuses on calculating the area under curves, among other things. For students, it's crucial to grasp integral calculus as it helps in understanding how functions behave and change over intervals. One practical method used in integral calculus is integration by parts. This technique is analogous to the product rule in differential calculus, involving two functions whose product's integral is sought. In essence, integral calculus is the fundamental step towards resolving complex calculus problems, and mastering it increases problem-solving skills significantly. Some common applications include solving problems related to areas, volumes, and central points.

Repeated Integration

Repeated integration refers to a technique where integration by parts is performed multiple times to simplify an integral further. In this process, a table is often used to organize the calculations efficiently. The integration by parts formula \[ \int u \, dv = uv - \int v \, du \]is applied iteratively until the integral becomes straightforward to manage. Understanding repeated integration is pivotal for tackling more complicated integrals like polynomial expressions combined with exponential or trigonometric functions. It allows students to methodically break down the problem into manageable steps, paving the way for a clearer and more structured problem-solving approach.

Function Selection

The success of integration by parts heavily depends on the appropriate selection of the functions \( u \) and \( dv \). Ideally, \( u \) should be a function that becomes simpler upon differentiation, whereas \( dv \) should be straightforward to integrate. This strategic choice simplifies the process, resulting in faster and more accurate solutions. In the context of \( \int x^2 e^{2x} \, dx \), choosing \( u = x^2 \) allows one to repeatedly reduce the power of \( x \), simplifying the integrals in each step. Understanding the logic behind function selection ensures not just solving the problem at hand but also instilling confidence in approaching different types of integrals.

Diagonal Product Method

The diagonal product method is an efficient strategy in repeated integration by parts when dealing with polynomials multiplied by exponential functions. By creating a table, each row represents a step where functions are either differentiated or integrated according to their columns. The diagonals of the table illustrate the products that need to be calculated. This method helps in visualizing and organizing the integral calculations in a streamlined fashion. Alternating signs are applied as per the integration by parts rule, ensuring the accuracy of the results. The diagonal product method, thus, transforms a potentially complex integration task into a systematic procedure that enhances clarity and precision.