Unit 03 · Week 8 · Lesson 06 · 60–90 minutes

Substitution for √(x²−a²)

Using x=a sec θ and careful domains. Build the idea, reconstruct the examples, select a pathway, test understanding, complete the printable college worksheet, and preserve a reflection in the local lesson record.

Begin LessonOpen Course Lab

Learning goals

Know the method, its conditions, and why it works.

Explain using x=a sec θ and careful domains.
Carry out a complete symbolic or numerical solution with conditions and units.
Compare at least two representations or methods.
Diagnose a plausible error and verify the repaired result.

x=a secθ√(x²−a²)=a tanθ

Seminar route

Retrieve prerequisite ideas and notation.
Analyze structure, hypotheses, and target quantity.
Rebuild both worked examples without copying.
Complete the six-part printable worksheet.
Correct, verify, reflect, and archive evidence.

Concept development

Build the structure before choosing the procedure.

Substitution for √(x²−a²) belongs to the broader Calculus II practice of classifying structure before computation. A complete solution addresses hypotheses, domain, exact notation, units, interpretation, and verification.

1 · Classify

Identify the mathematical structure and target quantity.

2 · Select

Choose a method whose hypotheses are satisfied.

3 · Execute

Show transformations with exact notation and units.

4 · Validate

Use an independent check and interpret the result.

Worked examples

Follow the reasoning one defensible step at a time.

Worked reasoning A

Problem: Evaluate ∫dx/√(x²−25).

Solution: ln|x+√(x²−25)|+C on |x|>5.

Worked reasoning B

Problem: Evaluate ∫√(x²−4)/x dx for x>2.

Solution: √(x²−4)−2arcsec(x/2)+C.

Pathway support

Choose scaffolding, standard college practice, or deeper extension.

Global and professional application

Use the mathematics in a defensible setting.

Design a concise model involving periodic energy, circular geometry, radical models, and exact measurement. Define variables and units, justify the mathematical object, compute or approximate a result, verify it independently, and state one limitation.

Portfolio expectation: retain one complete derivation, one error analysis, one interpreted application, and one independent verification.

Interactive quick check

Test method selection and reasoning before the worksheet.

These two questions save a local score only after both are answered. Study the explanation even when correct.

Khaemenes Higher Learning

Substitution for √(x²−a²)

Show method selection, derivation, conditions, units, verification, and interpretation.

U03 · L06

Name

Date

State the central method of Substitution for √(x²−a²) and list the conditions that must be checked.

Evaluate ∫dx/√(x²−25).

Evaluate ∫√(x²−4)/x dx for x>2.

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.

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.

Describe two independent ways to verify a result from Substitution for √(x²−a²).

Reflection and continuation

Explain what changed in your thinking.

What method, condition, error, or verification idea will you remember?

β Beta Program

Substitution for √(x²−a²)

Know the method, its conditions, and why it works.

Seminar route

Build the structure before choosing the procedure.

Follow the reasoning one defensible step at a time.

Worked reasoning A

Worked reasoning B

Choose scaffolding, standard college practice, or deeper extension.

Foundation pathway

Core college pathway

Honors / Extended pathway

Use the mathematics in a defensible setting.

Test method selection and reasoning before the worksheet.

Substitution for √(x²−a²)

Explain what changed in your thinking.