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

CAL2.24 Repeated Integration by Parts Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.24 Repeated Integration by Parts. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.24 Repeated Integration by Parts - Choose the correct antiderivative when integration by parts must be applied more than once. 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

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 CAL2.24 Repeated Integration by Parts skill assessments:

Problem 1Difficulty: Mastery

A rate is modeled by \(r(t)=(t² + t + 1) \sin(6t)\). Which accumulated-quantity formula \(A(t)\) has \(A'(t)=r(t)\), ignoring the initial constant?

A\(\frac{(2 t + 1) \sin(6t)}{36} + \frac{(- t² - t - 1) \cos(6t)}{6}\)

B\(\frac{(- t² - t - 1) \cos(6t)}{6}\)

C\(- \frac{(2 t + 1) \sin(6t)}{36} + \frac{(- t² - t - 1) \cos(6t)}{6} + \frac{\cos(6t)}{108}\)

D\(\frac{(2 t + 1) \sin(6t)}{36} - \frac{(t² + t + 1) \cos(6t)}{6} + \frac{\cos(6t)}{108}\)

Solution & Explanation:Repeated integration by parts reduces the polynomial factor and alternates sine and cosine terms, with each step adding another frequency denominator. Differentiating \(\frac{(2 t + 1) \sin(6t)}{36} - \frac{(t² + t + 1) \cos(6t)}{6} + \frac{\cos(6t)}{108}\) gives \((t² + t + 1) \sin(6t)\). The tempting errors are usually a lost sign or a missing power of the trig frequency.

Problem 2Difficulty: Mastery

A tabular integration setup repeatedly reduces one factor in \(\int (x² + 1)² e^-4x\,dx\). Which antiderivative is correct, ignoring \(C\)?

A\(- \frac{(32 x⁴ - 32 x³ + 88 x² - 44 x + 43) e^-4x}{128}\)

B\(- \frac{(32 x⁴ - 32 x³ + 88 x² - 20 x + 43) e^-4x}{128}\)

C\(- \frac{(32 x⁴ + 32 x³ + 88 x² + 44 x + 43) e^-4x}{128}\)

D\(- \frac{(32 x⁴ + 32 x³ + 88 x² + 44 x + 37) e^-4x}{128}\)

Solution & Explanation:Repeated integration by parts with the polynomial as \(u\) reduces the polynomial one derivative at a time; for \(p(x)e^ax\) the signs alternate as \(p/a-p'/a²+p''/a³-\cdots\). Differentiating \(- \frac{(32 x⁴ + 32 x³ + 88 x² + 44 x + 43) e^-4x}{128}\) gives \((x² + 1)² e^-4x\). A common mistake is to stop before the last polynomial-derivative term or to make all signs positive.

Problem 3Difficulty: Mastery

After applying integration by parts more than once to \(\int 2 x² (\ln x)⁵\,dx\), which final antiderivative is correct for x > 0?

A\(\frac{2 x³ (81 (\ln x)⁵ + 135 (\ln x)⁴ + 180 (\ln x)³ + 180 (\ln x)² + 120 \ln x + 40)}{243}\)

B\(\frac{2 x³ (81 (\ln x)⁵ - 135 (\ln x)⁴ + 180 (\ln x)³ - 180 (\ln x)² + 120 \ln x - 40)}{243}\)

C\(\frac{2 x³ (27 (\ln x)⁴ - 45 (\ln x)³ + 60 (\ln x)² - 60 \ln x + 40) \ln x}{81}\)

D\(\frac{2 x³ ((\ln x)⁵ - 5 (\ln x)⁴ + 20 (\ln x)³ - 60 (\ln x)² + 120 \ln x - 120)}{3}\)

Solution & Explanation:Let \(u=(\ln x)^m\) and \(dv=xⁿ dx\); each integration-by-parts step lowers the logarithm power by 1 until no log factor remains. Differentiating \(\frac{2 x³ (81 (\ln x)⁵ - 135 (\ln x)⁴ + 180 (\ln x)³ - 180 (\ln x)² + 120 \ln x - 40)}{243}\) gives \(2 x² (\ln x)⁵\). Stopping after only one reduction leaves an incomplete antiderivative.

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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 (Choose the correct antiderivative when integration by parts must be applied more than once. 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.).