Complete solutions, method expectations, validity conditions, and verification guidance.
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Khaemenes Higher Learning
Faculty Key · Reversing the Product Rule
Show method selection, derivation, conditions, units, verification, and interpretation.
U02 · L01
Name
Date
1
State the central method of Reversing the Product Rule and list the conditions that must be checked.
Faculty solution: Use deriving and applying integration by parts from the product rule. State the relevant hypotheses, domain or interval, units when applicable, and a verification plan.
2
Evaluate ∫x e^x dx.
Faculty solution: Take u=x and dv=e^x dx; the result is xe^x−e^x+C=e^x(x−1)+C.
3
Evaluate ∫x cos x dx.
Faculty solution: x sin x+cos x+C.
4
A learner makes an error by choosing u and dv without reducing complexity. Explain the first invalid step and repair the reasoning.
Faculty solution: The first invalid step is choosing u and dv without reducing complexity. Return to the governing definition or theorem, restore the missing condition, and verify the corrected result.
5
Create a short application of Reversing the Product Rule in work, moments, transform-like integrals, and repeated-product models. Define variables and identify the quantity an integral, differential equation, sequence, or series would measure.
Faculty solution: A defensible response defines variables and units, states assumptions, builds the correct mathematical object, and interprets its value in work, moments, transform-like integrals, and repeated-product models.
6
Describe two independent ways to verify a result from Reversing the Product Rule.
Faculty solution: Use an exact structural check such as differentiation, substitution, partial sums, or theorem hypotheses; then compare with numerical, graphical, dimensional, or limiting evidence.