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Khaemenes Higher Learning
Faculty Key · Substitution for √(x²−a²)
Show method selection, derivation, conditions, units, verification, and interpretation.
U03 · L06
Name
Date
1
State the central method of Substitution for √(x²−a²) and list the conditions that must be checked.
Faculty solution: Use using x=a sec θ and careful domains. State the relevant hypotheses, domain or interval, units when applicable, and a verification plan.
2
Evaluate ∫dx/√(x²−25).
Faculty solution: ln|x+√(x²−25)|+C on |x|>5.
3
Evaluate ∫√(x²−4)/x dx for x>2.
Faculty solution: √(x²−4)−2arcsec(x/2)+C.
4
A learner makes an error by using a trigonometric identity or substitution without matching the algebraic form. Explain the first invalid step and repair the reasoning.
Faculty solution: The first invalid step is using a trigonometric identity or substitution without matching the algebraic form. Return to the governing definition or theorem, restore the missing condition, and verify the corrected result.
5
Create a short application of Substitution for √(x²−a²) in periodic energy, circular geometry, radical models, and exact measurement. 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 periodic energy, circular geometry, radical models, and exact measurement.
6
Describe two independent ways to verify a result from Substitution for √(x²−a²).
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.