Solved: In the study of techniques of integration in calculus, certain indefinite

Chapter 3, Problem 64

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In the study of techniques of integration in calculus, certain indefinite integrals of the form \(\int e^{a x} f(x) d x\) could be evaluated by applying integration by parts twice, recovering the original integral on the right-hand side, solving for the original integral, and obtaining a constant multiple k \(\int e^{a x} f(x) d x\) on the left-hand side. Then the value of the integral is found by dividing by k. Discuss: For what kinds of functions f does the described procedure work? Your solution should lead to a differential equation. Carefully analyze this equation and solve for f.