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

CAL2.42 Integrals Requiring Multiple Techniques Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.42 Integrals Requiring Multiple Techniques. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.42 Integrals Requiring Multiple Techniques - Select the correct sequence of techniques for an integral that combines algebra, substitution, and parts. 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 CAL2.42 Integrals Requiring Multiple Techniques skill assessments:

Problem 1Difficulty: Mastery

Which sequence should be used for ∫ x²(ln(1+x³))² dx?

AUse parts first with dv = (ln(1+x³))² dx.

BLet u = ln(1+x³), then treat x^2dx as du/u without adjustment.

CUse partial fractions on 1+x³ before handling the logarithm.

DLet u = 1+x³, convert to a constant multiple of ∫ (ln u)² du, then use integration by parts, repeating once to finish.

Solution & Explanation:The factor x^2dx matches du/3 for u=1+x³, producing ∫(lnu)^2du. A logarithm squared integral is completed by parts, often using a second reduction step. Taking u as the logarithm changes du to include 1/(1+x³), which is not present.

Problem 2Difficulty: Mastery

The planned region sketch shows the area under y = x⁵ ln(1+x³) from x = 0 to x = 1. Which sequence computes the exact area?

AEstimate the area from the sketch and round.

BUse parts first with dv = ln(1+x³) dx.

CSet up ∫_0¹ x⁵ ln(1+x³) dx, let u = 1+x³ with bounds 1 to 2, rewrite x³ as u-1, then use integration by parts on polynomial times ln u.

DUse partial fractions on 1+x³ before setting up the area integral.

Solution & Explanation:The sketch provides the area integral, and substitution converts the extra power to u-1 with bounds 1 to 2. Polynomial-logarithm terms are then evaluated by parts. A rounded graph estimate is not exact.

Problem 3Difficulty: Mastery

For ∫ x⁵ ln(1+x³) dx, which method sequence is most efficient?

AUse parts first with dv = ln(1+x³) dx.

BLet u = 1+x³, rewrite x³ as u-1, expand, then use integration by parts on polynomial times ln u.

CLet u = ln(1+x³), which removes all powers of x.

DUse partial fractions after factoring x³+1.

Solution & Explanation:Writing x^5dx as x^3x^2dx makes u=1+x³ work. The leftover x³ becomes u-1, and the product with ln u is handled by parts. Choosing the logarithm as u leaves a nontrivial leftover factor.

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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 sequence of techniques for an integral that combines algebra, substitution, and parts. 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.).