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Khaemenes Higher Learning
Faculty Key · Definite Integration by Parts
Show method selection, derivation, conditions, units, verification, and interpretation.
U02 · L04
Name
Date
1
State the central method of Definite Integration by Parts and list the conditions that must be checked.
Faculty solution: Use tracking boundary terms and exact values in definite products. State the relevant hypotheses, domain or interval, units when applicable, and a verification plan.
2
Evaluate ∫₀¹ x e^x dx.
Faculty solution: [e^x(x−1)]₀¹=1.
3
Evaluate ∫₀^π x sin x dx.
Faculty solution: [−x cos x+sin x]₀^π=π.
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 Definite Integration by Parts 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 Definite Integration by Parts.
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.