Free Printable WorksheetMathCalculus IIMath CurriculumAP Calculus BC Unit 6; FL MA.912.C.4.*; CCSS.MP1, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-INT; TEKS 19 TAC section 111.55

CAL2.29 Trig Substitution for a^2 x^2 Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.29 Trig Substitution for a^2 x^2. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.29 Trig Substitution for a^2 - x^2 - Select the correct substitution, triangle relationship, and domain restriction for radicals of the form sqrt(a^2 - x^2). Standards: AP Calculus BC Unit 6; FL MA.912.C.4.*; CCSS.MP1, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-INT; TEKS 19 TAC section 111.55

Whether you are an educator preparing classroom handouts, a tutor reinforcing weekly topics, or a parent supporting homeschool learning, this printable resource gives students the deliberate practice required to achieve subject fluency and master state standards.

Grade

Calculus II

Subject

Math

Domain

Math Curriculum

Formats

PDF & DOCX

Sample Practice Problems

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Below are sample questions representative of the exercises included on this worksheet and related CAL2.29 Trig Substitution for a^2 x^2 skill assessments:

Problem 1Difficulty: Mastery
For ∫ (x - 2)2/sqrt(25 - (x - 2)2) dx, which complete trig-substitution setup is correct?
Ax = 5 sin(theta); transformed integrand = 25 sin2(theta) dtheta
Bx - 2 = 5 tan(theta); transformed integrand = 25 tan2(theta) dtheta
Cx - 2 = 25 sin(theta); transformed integrand = 625 sin2(theta) dtheta
Dx - 2 = 5 sin(theta); transformed integrand = 25 sin2(theta) dtheta
Solution & Explanation:Let u = x - 2 = 5 sin(theta). Then u2 = 25 sin2(theta), dx = 5 cos(theta) dtheta, and the radical is 5 cos(theta), so the cosine factors cancel.
Problem 2Difficulty: Mastery
Under x = a sin(theta), what theta limits correspond to -a/2 <= x <= a/2?
A-pi/3 <= theta <= pi/3
B0 <= theta <= pi/3
C-pi/6 <= theta <= pi/6
D5pi/6 <= theta <= 7pi/6
Solution & Explanation:Dividing by a gives -1/2 <= sin(theta) <= 1/2. On the standard interval [-pi/2, pi/2], that corresponds to -pi/6 <= theta <= pi/6.
Problem 3Difficulty: Mastery
For sqrt(a2 - x2) with x = a sin(theta), which statement is correct at the endpoint x = a?
Atheta = 0, cos(theta) = 1, and the radical is a
Btheta = pi/2, cos(theta) = 0, and the radical is 0
Ctheta = -pi/2, cos(theta) = 0, and the radical is a
Dtheta = pi, cos(theta) = -1, and the radical is -a
Solution & Explanation:At x = a, sin(theta) = 1 on the standard interval, so theta = pi/2. The adjacent side and the radical both equal 0.

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QWhat grade level and standards does this worksheet align with?

This worksheet is designed specifically for Calculus II Math students. It aligns with educational standard AP Calculus BC Unit 6; FL MA.912.C.4.*; CCSS.MP1, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-INT; TEKS 19 TAC section 111.55 (Select the correct substitution, triangle relationship, and domain restriction for radicals of the form sqrt(a^2 - x^2). Standards: AP Calculus BC Unit 6; FL MA.912.C.4.*; CCSS.MP1, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-INT; TEKS 19 TAC section 111.55).

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