77. Tabular integration Consider the integral ∫ f(x)g(x) dx, where f can be differentiated repeatedly and g can be integrated repeatedly
Let Gₖ represent the result of calculating k indefinite integrals of g (omitting constants of integration).
d. The tabular integration table from part (c) is easily extended to allow for as many steps as necessary in the process of integration by parts.
Evaluate ∫ x² e^(x/2) dx by constructing an appropriate table, and explain why the process terminates after four rows of the table have been filled in.

Verified step by step guidance

1. Identify the functions for integration by parts: let \( f(x) = x^2 \) (which can be differentiated repeatedly) and \( g(x) = e^{x/2} \) (which can be integrated repeatedly).
2. Construct a table with two columns: one for successive derivatives of \( f(x) \) and one for successive integrals of \( g(x) \). Start by listing \( f(x) \) and its derivatives down the left column, and \( g(x) \) and its integrals down the right column.
3. Calculate the derivatives of \( f(x) = x^2 \): \( f'(x) = 2x \), \( f''(x) = 2 \), and \( f'''(x) = 0 \). This shows the differentiation process will terminate after the third derivative because the derivative becomes zero.
4. Calculate the integrals of \( g(x) = e^{x/2} \): the first integral is \( G_1 = \int e^{x/2} dx \), the second integral \( G_2 = \int G_1 dx \), and so on, each time integrating the previous result.
5. Use the tabular integration method by multiplying diagonally and alternating signs (+, -, +, -) to write the sum of products of derivatives of \( f \) and integrals of \( g \). The process terminates after four rows because the derivative of \( f \) becomes zero, so no further terms contribute to the integral.

Key Concepts

Integration by Parts

Integration by parts is a technique derived from the product rule of differentiation. It transforms the integral of a product of functions into simpler integrals, using the formula ∫u dv = uv - ∫v du. Choosing u and dv wisely simplifies the problem, especially when one function becomes simpler upon differentiation.

Tabular Integration Method

Tabular integration is a streamlined approach to repeated integration by parts. It organizes derivatives of one function and integrals of the other in a table, alternating signs to quickly sum terms. This method is efficient when one function differentiates to zero after finite steps, allowing the process to terminate naturally.

Termination of the Integration Process

The integration process terminates when repeated differentiation of one function results in zero, eliminating further terms. In the example ∫ x² e^(x/2) dx, differentiating x² repeatedly leads to zero after the third derivative, so the tabular method ends after four rows, ensuring a finite and complete solution.