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

CAL2.25 Definite 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 CAL2.25 Definite Integration by Parts. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.25 Definite Integration by Parts - Select the correct definite-integral value using integration by parts and boundary evaluation. Standards tags: 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 CAL2.25 Definite Integration by Parts skill assessments:

Problem 1Difficulty: Mastery

For a nonzero constant \(a\), select the value of \(\int_0¹ x\cos(ax)\,dx\) using integration by parts.

A\(\frac{\sin a}{a}+\frac{1-\cos a}{a²}\)

B\(\frac{\cos a}{a}+\frac{\sin a}{a²}\)

C\(\frac{\sin a}{a}-\frac{\cos a}{a²}\)

D\(\frac{\sin a}{a}+\frac{\cos a-1}{a²}\)

Solution & Explanation:Take \(u=x\) and \(dv=\cos(ax)\,dx\), so \(v=\sin(ax)/a\). Then \(\int_0¹ x\cos(ax)\,dx=[x\sin(ax)/a]_0¹-\frac{1}{a}\int_0¹\sin(ax)\,dx\), which is \(\frac{\sin a}{a}+\frac{\cos a-1}{a²}\). The sign of the cosine boundary term is the usual trap. The correct choice is D.

Problem 2Difficulty: Mastery

Let \(f\) be differentiable on \([0,1]\), with \(f(0)=2\), \(f(1)=5\), \(\int_0¹ f(x)\,dx=3\), and \(\int_0¹ x f(x)\,dx=1\). Evaluate \(\int_0¹ x(1-x)f'(x)\,dx\).

A\(1\)

B\(0\)

C\(-1\)

D\(-3\)

Solution & Explanation:Use \(u=x(1-x)\), \(dv=f'(x)\,dx\). The boundary term is 0 because \(x(1-x)=0\) at both endpoints. Then subtract \(\int_0¹ (1-2x)f(x)\,dx=3-2(1)=1\), giving \(-1\). The common mistake is to use \(+1\) instead of subtracting. The correct choice is C.

Problem 3Difficulty: Mastery

For an integer \(n\ge 0\), select the value of the convergent improper integral \(I_n=\int_0¹ xⁿ\ln x\,dx\) found by integration by parts.

A\(\frac{1}{(n+1)²}\)

B\(-\frac{1}{(n+1)²}\)

C\(-\frac{1}{n+1}\)

D\(-\frac{1}{n²+1}\)

Solution & Explanation:Take \(u=\ln x\) and \(dv=xⁿ\,dx\). The boundary term \(\left[\frac{xⁿ⁺¹\ln x}{n+1}\right]_0¹\) is 0 by the one-sided limit at 0, and the remaining integral is \(-\frac{1}{n+1}\int_0¹ xⁿ\,dx=-\frac{1}{(n+1)²}\). The sign is often lost when subtracting \(v\,du\). The correct choice is B.

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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 definite-integral value using integration by parts and boundary evaluation. Standards tags: 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).