Free Printable WorksheetMathCalculus IIMath CurriculumAP Calculus BC Unit 10; CCSS.HSA.SSE, HSF.IF; CCSS.MP2, MP3, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-SER; TEKS 19 TAC section 111.55

CAL2.48 p Integral 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 CAL2.48 p Integral Test for Improper Integrals. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.120 Integral Test - Choose the correct integral-test conclusion after verifying positivity, continuity, and decreasing behavior. Tags: AP Calculus BC Unit 10; CCSS.HSA.SSE, HSF.IF; CCSS.MP2, MP3, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-SER; 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 CAL2.48 p Integral Test for Improper Integrals skill assessments:

Problem 1Difficulty: Mastery

A student wants to apply the Integral Test directly to ∑_{n=1}^∞ sin n/n² using f(x)=sin x/x². Which statement is correct?

AThe series converges by the Integral Test because the terms are bounded.

BThe series diverges by the Integral Test because the terms are not all zero.

CThe Integral Test applies without checking positivity, continuity, or decreasing behavior.

DThe Integral Test is not justified as stated because the proposed f is not positive and decreasing on a tail interval.

Solution & Explanation:The Integral Test requires a positive, continuous, decreasing function f with a_n=f(n) eventually. The sine factor changes sign and oscillates, so the direct Integral Test hypotheses fail. Other tests may show convergence, but this proposed Integral Test argument is not valid. The test may not be used merely because an expression resembles an improper integral.

Problem 2Difficulty: Mastery

Apply the Integral Test to ∑_{n=1}^∞ n³/(n⁴+1). The matching function is positive, continuous, and decreasing on a tail interval. Which conclusion is correct?

AThe series converges by the Integral Test because ∫_1^∞ x³/(x⁴+1) dx diverges.

BThe Integral Test proves convergence whenever the terms approach 0.

CThe series diverges by the Integral Test because ∫_1^∞ x³/(x⁴+1) dx diverges.

DThe Integral Test cannot be used because the terms are decreasing.

Solution & Explanation:For the stated function, the positivity, continuity, and decreasing hypotheses hold on the stated interval or on a tail interval. The substitution u=x⁴+1 gives a logarithmic integral, so the tail is unbounded. A common mistake is to reverse the integral conclusion or to ignore one of the Integral Test hypotheses.

Problem 3Difficulty: Mastery

For ∑_{n=2}^∞ n³/(n⁵+1), the matching function is positive, continuous, and decreasing for x≥ 2. What follows by the Integral Test?

AThe series diverges by the Integral Test because ∫_2^∞ x³/(x⁵+1) dx converges.

BThe series converges by the Integral Test because ∫_2^∞ x³/(x⁵+1) dx converges.

CThe Integral Test is inconclusive because the terms approach 0.

DThe Integral Test cannot be used because the terms are positive.

Solution & Explanation:For the stated function, the positivity, continuity, and decreasing hypotheses hold on the stated interval or on a tail interval. For x≥ 2, the integrand is bounded above by 1/x² up to a constant-size tail comparison, so the improper integral is finite. A common mistake is to reverse the integral conclusion or to ignore one of the Integral Test hypotheses.

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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 10; CCSS.HSA.SSE, HSF.IF; CCSS.MP2, MP3, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-SER; TEKS 19 TAC section 111.55 (Choose the correct integral-test conclusion after verifying positivity, continuity, and decreasing behavior. Tags: AP Calculus BC Unit 10; CCSS.HSA.SSE, HSF.IF; CCSS.MP2, MP3, MP6, MP7, MP8; Integrated Calculus II Standards CAL2-SER; TEKS 19 TAC section 111.55).