Free Printable WorksheetMathCalculus IIMath CurriculumAP Calculus BC Unit 6 and Unit 8; FL MA.912.C.4.6, C.4.8-C.4.10; CCSS.MP2, MP3, MP5, MP6; Integrated Calculus II Standards CAL2-IMP, CAL2-NUM; TEKS 19 TAC section 111.55

CAL2.44 Improper Integrals with Infinite Limits Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.44 Improper Integrals with Infinite Limits. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.44 Improper Integrals with Infinite Limits - Select the correct limit expression and convergence conclusion for an integral over an infinite interval. Aligns with AP Calculus BC Unit 6 and Unit 8; FL MA.912.C.4.6, C.4.8-C.4.10; CCSS.MP2, MP3, MP5, MP6; Integrated Calculus II Standards CAL2-IMP, CAL2-NUM; 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 CAL2.44 Improper Integrals with Infinite Limits skill assessments:

Problem 1Difficulty: Mastery

Which choice has both the correct limit setup and convergence conclusion for ∫_1^∞ 1/[x(1+ln x)] dx?

Alim_{b→∞} ∫_1^b 1/[x(1+ln x)] dx = 1; converges

Blim_{b→∞} ∫_1^b 1/[x(1+ln x)] dx = 0; converges

Clim_{b→1} ∫_1^b 1/[x(1+ln x)] dx = 0; converges

Dlim_{b→∞} ∫_1^b 1/[x(1+ln x)] dx = lim_{b→∞} ln(1+ln b) = ∞; diverges

Solution & Explanation:Use u=1+ln x. Then the integral is ∫_1^{1+ln b}du/u=ln(1+ln b), which grows without bound. This is the shifted logarithmic borderline case.

Problem 2Difficulty: Mastery

Use the improper-integral definition for ∫_1^∞ 1/[x(1+ln x)²] dx. Which conclusion is correct?

Alim_{b→∞} ∫_1^b 1/[x(1+ln x)²] dx = ∞; diverges

Blim_{b→∞} ∫_1^b 1/[x(1+ln x)²] dx = ln 2; converges

Clim_{b→∞} ∫_1^b 1/[x(1+ln x)²] dx = lim_{b→∞}(1 - 1/(1+ln b)) = 1; converges

Dlim_{b→1} ∫_1^b 1/[x(1+ln x)²] dx = 0; diverges

Solution & Explanation:Let u=1+ln x, so du=dx/x and u runs from 1 to 1+ln b. The integral is ∫_1^{1+ln b}u^-2du=1−1/(1+ln b), which tends to 1.

Problem 3Difficulty: Mastery

Which option correctly defines ∫_1^∞ 1/[x(1+√(ln x))] dx with a limit and states convergence or divergence?

Alim_{b→∞} ∫_1^b 1/[x(1+√(ln x))] dx converges because √(ln x) grows

Blim_{b→∞} ∫_1^b 1/[x(1+√(ln x))] dx diverges; with u=√(ln x), the transformed integral grows without bound

Clim_{b→∞} ∫_1^b 1/[x(1+√(ln x))] dx converges by comparison with 1/x²

Dlim_{b→1} ∫_1^b 1/[x(1+√(ln x))] dx = 0; this proves convergence

Solution & Explanation:Let u=√(ln x), so dx/x=2u du. The transformed integrand is 2u/(1+u), whose integral grows like 2u. Therefore the improper integral diverges.

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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 and Unit 8; FL MA.912.C.4.6, C.4.8-C.4.10; CCSS.MP2, MP3, MP5, MP6; Integrated Calculus II Standards CAL2-IMP, CAL2-NUM; TEKS 19 TAC section 111.55 (Select the correct limit expression and convergence conclusion for an integral over an infinite interval. Aligns with AP Calculus BC Unit 6 and Unit 8; FL MA.912.C.4.6, C.4.8-C.4.10; CCSS.MP2, MP3, MP5, MP6; Integrated Calculus II Standards CAL2-IMP, CAL2-NUM; TEKS 19 TAC section 111.55.).