Question

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.

Accepted Answer

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). 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. 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. 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. 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.

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Key Concepts

Integration by Parts

Tabular Integration Method

Termination of the Integration Process