Problem 85 - 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. \(\int x^{3} e^{2 x} d x\)

Step-by-Step Solution

Answer

The integral is \( \frac{1}{2}x^3 e^{2x} - \frac{3}{4}x^2 e^{2x} + \frac{3}{4}x e^{2x} - \frac{3}{8} e^{2x} + C \).

Step 1: Choose Functions for Integration by Parts

We choose functions for integration by parts. Let \( u = x^3 \) and \( dv = e^{2x} \, dx \). Then, compute \( du \) and \( v \) as follows: - \( du = 3x^2 \, dx \)- Integrate \( dv \) to find \( v = \frac{1}{2}e^{2x} \)

Step 2: Set Up the Integration by Parts Table

Create a table with three columns for \( u \), \( dv \), and the derivatives and integrals:| \( u \) | \( dv \) ||----------------|------------------|| \( x^3 \) | \( e^{2x} \) || \( 3x^2 \) | \( \frac{1}{2}e^{2x} \) || \( 6x \) | \( \frac{1}{4}e^{2x} \) || \( 6 \) | \( \frac{1}{8}e^{2x} \) || \( 0 \) | \( \frac{1}{16}e^{2x} \) |

Step 3: Apply Diagonal Cross-Multiplication

We find the integral by multiplying diagonally across the table and paying attention to signs:1. \( x^3 \times \frac{1}{2}e^{2x} = \frac{1}{2}x^3 e^{2x} \)2. \(- (3x^2 \times \frac{1}{4}e^{2x}) = -\frac{3}{4}x^2 e^{2x} \)3. \(+ (6x \times \frac{1}{8}e^{2x}) = \frac{3}{4}x e^{2x} \)4. \(- (6 \times \frac{1}{16}e^{2x}) = -\frac{3}{8} e^{2x} \)

Step 4: Summarize the Result with the Constant of Integration

Add all the products from Step 3 together and include the constant of integration \( C \):\[ \int x^3 e^{2x} \, dx = \frac{1}{2}x^3 e^{2x} - \frac{3}{4}x^2 e^{2x} + \frac{3}{4}x e^{2x} - \frac{3}{8} e^{2x} + C \]

Key Concepts

Integration Techniques

Integration is a fundamental concept in calculus that is often used to find the area under a curve. However, not all integrals can be computed directly using basic techniques like substitution. This is where specialized integration techniques, such as integration by parts, become valuable. Integration by parts is based on the product rule for differentiation and is typically used when the integral involves a product of two functions.
In its essence, the integration by parts formula is: \[ \int u \ dv = uv - \int v \ du \] This technique requires selecting parts of the integral to represent as \( u \) and \( dv \), while \( du \) is the derivative of \( u \), and \( v \) is the antiderivative of \( dv \). Choosing \( u \) and \( dv \) correctly is crucial for simplifying the problem efficiently.
Repeated application of integration by parts can be systematically organized using tables, which helps visualize the steps and reduce errors in calculation. It makes life easier when dealing with integrals that seem complex at first glance.

Integration by Parts Table

The integration by parts table is a structured method that simplifies the repeated application of the integration by parts technique. This table consists of rows where one column lists the function \( u \) and its derivatives, while another column lists the function \( dv \) and its integrals. This approach simplifies tracking the calculations and reducing chances of errors during the process.
Creating a table for the integral \( \int x^3 e^{2x} \, dx \) involves listing the derivatives of \( x^3 \) in one column and the integrals of \( e^{2x} \, dx \) in another. As you progress down the rows:

• An initial row contains \( u = x^3 \) and \( dv = e^{2x} \, dx \).
• The functions \( du \) and \( v \) are entered in the subsequent rows by taking derivatives and antiderivatives respectively.

The table method also aids in implementing the diagonal cross multiplication process to combine results.
By carefully managing signs and performing cross multiplications, each step involves creating smaller components of the integral result, which are later summed to obtain the final integral expression.

Calculus Problem Solving

Calculus problem solving is a skill that involves applying various methods and strategies to solve complex mathematical problems. When faced with integrals that are not straightforward, as in the exercise of finding \( \int x^3 e^{2x} \, dx \), integrating effectively can involve multiple calculus techniques.
It's crucial to recognize the type of functions involved and choose the appropriate technique, such as integration by parts. Solving these problems requires a keen understanding of calculus rules and formulas, as well as the ability to use shortcuts and techniques, such as table methods, to reduce computation complexity.
During problem solving:

• Carefully choose \( u \) and \( dv \) parts for integration by parts.
• Organize the problem-solving process to minimize mistakes.
• Use constants of integration where necessary.

Through practice, one improves at recognizing which techniques to use and when. Thus, calculus problem solving becomes not just a mechanical computation but an art of logical reasoning and structuring to reach the correct solution.