Problem 83 - 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. \(\int x^{2} e^{-x} d x\)

Step-by-Step Solution

Answer

\( \int x^2 e^{-x} \, dx = -x^2 e^{-x} - 2x e^{-x} - 2e^{-x} + C \)

Step 1: Choose Functions for Integration by Parts

In integration by parts, we choose \( u \) and \( dv \). For our problem, let \( u = x^2 \) so that \( du = 2x \, dx \), and let \( dv = e^{-x} \, dx \) so that \( v = -e^{-x} \).

Step 2: Set Up the Table

Create a table with three columns to organize the process. The left column represents the derivatives of \( u \), and the middle column represents the integrals of \( dv \). The third column will be used to indicate the sign for each diagonal product:\( \begin{array}{c|c|c}\text{Derivatives of } u & \text{Integrals of } dv & \text{Sign} \hlinex^2 & -e^{-x} & + \hline2x & e^{-x} & - \hline2 & -e^{-x} & + \hline0 & e^{-x} & \\end{array} \)

Step 3: Multiply Diagonally and Apply Signs

Multiply diagonally and apply the corresponding sign to find each term of the integral:- First diagonal: \( x^2 \cdot (-e^{-x}) = -x^2 e^{-x} \) with positive sign.- Second diagonal: \( 2x \cdot e^{-x} = 2x e^{-x} \) with negative sign.- Third diagonal: \( 2 \cdot (-e^{-x}) = -2e^{-x} \) with positive sign.

Step 4: Write Out the Integral Result

Combine the products from the diagonals, adding a constant of integration \( C \):\[\int x^2 e^{-x} \, dx = -x^2 e^{-x} - 2x e^{-x} - 2e^{-x} + C\]

Key Concepts

Repeated Integration

Repeated integration, particularly through the method of integration by parts, is a powerful technique when dealing with the product of two functions. The integration by parts formula is derived from the product rule for differentiation and can be expressed as:\[\int u \, dv = uv - \int v \, du\]In many instances, however, a single application of integration by parts does not solve the problem entirely, this is where repeated integration by parts becomes essential.
To streamline this process, especially for problems like integrating \(\int x^2 e^{-x} \, dx\), a table method can be employed. This method not only organizes all derivatives and integrals but also clarifies where signs should be positive or negative. By consistently applying integration by parts, new terms emerge, eventually simplifying enough to solve the integral completely.

Integration Techniques

Integration requires a variety of techniques depending on the nature of the functions involved. When dealing with products involving polynomials and exponential functions, integration by parts is often the go-to method. This approach is particularly useful when trying to reduce the power of the polynomial.
Setting up an efficient strategy using integration by parts involves:

• Choosing functions 'u' and 'dv' wisely to simplify the problem.
• Using a table to organize derivatives and integrals systematically.
• Ensuring proper handling of signs as dictated by diagonal multiplication in the table.

Repetitive practice of this technique boosts intuition for selecting the right parts and getting comfortable with the process of repeated integration by parts. The aim is to reduce a complex integral into simpler parts that are easy to handle or recognizable.

Exponential Functions

Exponential functions, such as \(e^{x}\) or \(e^{-x}\), play a critical role in integration since their unique properties make them relatively straightforward to integrate and differentiate. In the exercise \(\int x^2 e^{-x} \, dx\), it's clear that rubbing shoulders with a polynomial, the exponential function's derivative remains unchanged in form aside from losing the coefficient.
Integrating an exponential function results in another exponential function, and when combined with a polynomial, they create necessitated interactions for integration techniques. As these functions have constant growth rates, they make integration processes such as by parts, manageable and intuitive, especially after setting up the appropriate u and dv. This highlights one of the integral strengths of integration by parts: the ability to systematically manage functions whose derivatives and integrals create cascading effects useful for simplifying expressions.