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

CAL2.27 Trigonometric Integrals with Powers of Tangent and Secant Worksheet

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Worksheet Overview & Skills Covered

Core academic competencies and curriculum alignment

This Calculus II Math worksheet provides structured practice on CAL2.27 Trigonometric Integrals with Powers of Tangent and Secant. Students are guided through targeted exercises that build foundational understanding, computational accuracy, and critical thinking skills.

CAL2.27 Trigonometric Integrals with Powers of Tangent and Secant - Select the correct rewrite for integrals involving powers of tangent and secant. 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 CAL2.27 Trigonometric Integrals with Powers of Tangent and Secant skill assessments:

Problem 1Difficulty: Mastery

For ∫ 3tan⁵(π/4-x/2) sec⁵(π/4-x/2) dx, choose the rewrite that keeps the substitution u=sec(π/4-x/2) consistent.

A\(-6\int u⁴(1+u²)²\,du\)

B\(-6\int u⁵(u²-1)²\,du\)

C\(-6\int u⁴(u²-1)³\,du\)

D\(-6\int u⁴(u²-1)²\,du\)

Solution & Explanation:Because the tangent power is odd, save a sec(π/4-x/2) tan(π/4-x/2) dx factor. With u=sec(π/4-x/2), du=-(1)/(2) sec(π/4-x/2) tan(π/4-x/2) dx, so the outside factor is -6. The remaining tangent power becomes (u²-1)². The rewrite is -6∫ u⁴(u²-1)² du. A common mistake is to change tangent powers using 1+u²; that identity is for u=tan, not u=sec.

Problem 2Difficulty: Mastery

After saving the differential factor for u=tan(π/6-x/3), which rewrite is correct for ∫ 2tan²(π/6-x/3) sec⁴(π/6-x/3) dx?

A\(-6\int u²(u²-1)\,du\)

B\(-6\int u²(1+u²)²\,du\)

C\(-6\int u²(1+u²)\,du\)

D\(-6\int u⁴(1+u²)\,du\)

Solution & Explanation:Because the secant power is even, save a sec²(π/6-x/3) dx factor. With u=tan(π/6-x/3), du=-(1)/(3)sec²(π/6-x/3) dx, so the outside factor is -6. The remaining secant power becomes (1+u²). The rewrite is -6∫ u²(1+u²) du. A common mistake is to use u²-1, which comes from tan²=sec²-1 and belongs to a u=sec rewrite.

Problem 3Difficulty: Mastery

Use u=sec(2x+π/6). Which expression is the equivalent u-integral for ∫ -6tan⁷(2x+π/6) sec³(2x+π/6) dx?

A\(-3\int u²(1+u²)³\,du\)

B\(-3\int u²(u²-1)³\,du\)

C\(-3\int u³(u²-1)³\,du\)

D\(-3\int u²(u²-1)⁴\,du\)

Solution & Explanation:Because the tangent power is odd, save a sec(2x+π/6) tan(2x+π/6) dx factor. With u=sec(2x+π/6), du=2 sec(2x+π/6) tan(2x+π/6) dx, so the outside factor is -3. The remaining tangent power becomes (u²-1)³. The rewrite is -3∫ u²(u²-1)³ du. A common mistake is to change tangent powers using 1+u²; that identity is for u=tan, not u=sec.

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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 rewrite for integrals involving powers of tangent and secant. 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).