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

CAL2.49 Comparison Test for Improper Integrals Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.49 Comparison Test for Improper Integrals. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.49 Comparison Test for Improper Integrals - Choose the valid comparison function and conclusion for a nonnegative improper integrand. Standards tags: 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; CAL2-IMP; CAL2-NUM; TEKS 19 TAC section 111.55. (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

Preview the types of problems and questions covered in this worksheet

Below are sample questions representative of the exercises included on this worksheet and related CAL2.49 Comparison Test for Improper Integrals skill assessments:

Problem 1Difficulty: Mastery

Consider \(I=\int_0¹ \frac{x^1/2+x}{x^5/4+x^3/2}\,dx\), with integrand \(f(x)\). Which option gives a valid direct comparison for the nonnegative integrand and the correct conclusion?

AFor \(03/4}\le f(x)\); since \(\int_0¹ \frac{2}{x^3/4}\,dx\) converges, \(I\) converges.

BFor \(01 \frac{1}{x}\,dx\) diverges, \(I\) diverges.

CFor \(03/4}\); since \(\int_0¹ \frac{2}{x^3/4}\,dx\) converges, \(I\) diverges.

DFor \(03/4}\); since \(\int_0¹ \frac{2}{x^3/4}\,dx\) converges, \(I\) converges.

Solution & Explanation:Correct answer: D. For \(01/2+x\le2x^1/2\) and the denominator is at least \(x^5/4\), so \(f(x)\le2/x^3/4\). The comparison function has a convergent improper integral, so the direct comparison test proves that \(I\) converges. An upper bound by a divergent integral, or a lower bound by a convergent integral, would not decide convergence.

Problem 2Difficulty: Mastery

Consider \(I=\int_0¹ \frac{x^1/2+x}{x^3/2+x²}\,dx\), with integrand \(f(x)\). Which option gives a valid direct comparison for the nonnegative integrand and the correct conclusion?

AFor \(01 \frac{1}{2x}\,dx\) diverges, \(I\) diverges.

BFor \(01 \frac{1}{\sqrt{x}}\,dx\) converges, \(I\) converges.

CFor \(01 \frac{1}{2x}\,dx\) diverges, \(I\) diverges.

DFor \(01 \frac{1}{2x}\,dx\) diverges, \(I\) converges.

Solution & Explanation:Correct answer: C. The numerator is at least \(x^1/2\), while \(x^3/2+x²\le2x^3/2\), so \(f(x)\ge1/(2x)\). The comparison function has a divergent improper integral and lies below the nonnegative integrand, so the direct comparison test proves that \(I\) diverges. An upper divergent bound or a lower convergent bound is a common but invalid shortcut.

Problem 3Difficulty: Mastery

Consider \(I=\int_1^\infty \frac{x²+\ln x}{x³+x}\,dx\), with integrand \(f(x)\). Which option gives a valid direct comparison for the nonnegative integrand and the correct conclusion?

AFor \(x\ge 1\), \(0\le f(x)\le \frac{1}{2x}\); since \(\int_1^\infty \frac{1}{2x}\,dx\) diverges, \(I\) diverges.

BFor \(x\ge 1\), \(0\le \frac{1}{2x}\le f(x)\); since \(\int_1^\infty \frac{1}{2x}\,dx\) diverges, \(I\) diverges.

CFor \(x\ge 1\), \(0\le \frac{1}{x²}\le f(x)\); since \(\int_1^\infty \frac{1}{x²}\,dx\) converges, \(I\) converges.

DFor \(x\ge 1\), \(0\le \frac{1}{2x}\le f(x)\); since \(\int_1^\infty \frac{1}{2x}\,dx\) diverges, \(I\) converges.

Solution & Explanation:Correct answer: B. The numerator is at least \(x²\), and \(x³+x\le2x³\), so \(f(x)\ge1/(2x)\). The comparison function has a divergent improper integral and lies below the nonnegative integrand, so the direct comparison test proves that \(I\) diverges. An upper divergent bound or a lower convergent bound is a common but invalid shortcut.

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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 (Choose the valid comparison function and conclusion for a nonnegative improper integrand. Standards tags: 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; CAL2-IMP; CAL2-NUM; TEKS 19 TAC section 111.55.).