Free printable worksheet

u-Substitution Worksheet

Twelve substitution integrals with algebraic, exponential, logarithmic, and trigonometric forms.

CalculusGrades 11-124 pagesAnswer key included

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What this worksheet includes

  • Twelve substitution integrals with algebraic, exponential, logarithmic, and trigonometric forms.
  • 4-page printable PDF with a complete answer key
  • Designed for Grades 11-12 calculus practice
  • First-page preview available before the free email unlock

How to use it

Use this u-substitution worksheet for guided practice, independent review, homework, tutoring sessions, or a quick skills check. The spacious layout is designed to leave room for students to show their work rather than guess from crowded answer choices.

After students finish, compare their reasoning with the included answer key. When an answer is incorrect, have the student locate the first step where their work changed direction before trying the problem again.

Who this practice is for

This resource is designed for grades 11-12 students working on calculus. Teachers, tutors, homeschool families, and parents can preview the first page before deciding whether the level and spacing fit the student.

Looking for a broader sequence? Browse all Calculus worksheets.

Topic guide

How to approach u-substitution

u-Substitution practice is most effective when students identify the governing definition, theorem, or rule before beginning a calculation. Twelve substitution integrals with algebraic, exponential, logarithmic, and trigonometric forms.

This calculus set moves from direct applications into mixed review. Students should preserve exact values, label each important step, and use the answer key to locate the first point where an incorrect solution changes direction.

Learning objectives

  • Recognize the central rules and representations used in u-substitution
  • Apply an efficient procedure to solve u-substitution problems
  • Check results using notation, substitution, estimation, or a relevant inverse process

Skills to review first

  • Core vocabulary connected to u-substitution
  • Accurate arithmetic and algebraic simplification
  • Reading mathematical notation and showing work in a logical sequence
Worked example

Follow the reasoning step by step

Plan and solve a representative u-substitution problem from Part A.

  1. List the given information and state what must be found.
  2. Select the definition, theorem, formula, or procedure that applies to u-substitution.
  3. Substitute or transform carefully, keeping one logical change on each line.
  4. Check the result against the original conditions and include units or restrictions when needed.

Solution: A complete solution includes the correct result and enough reasoning to verify it.

Try before downloading

Three sample problems

Work each problem, then open the answer to check your result.

Problem 1 · foundations

Evaluate using u-substitution: ∫2x(x2+1)4 dx\int2x(x^2+1)^4\,dx

Show answer

(x2+1)55+C\frac{(x^2+1)^5}{5}+C

Problem 2 · skill practice

Evaluate using u-substitution: ∫13x+1 dx\int\frac{1}{3x+1}\,dx

Show answer

13ln⁡∣3x+1∣+C\frac13\ln|3x+1|+C

Problem 3 · mixed review

Evaluate using u-substitution: ∫0π/2sin⁡xcos⁡x dx\int_0^{\pi/2}\sin x\cos x\,dx

Show answer

12\frac12

Error check

Common mistakes to watch for

  • Beginning a calculation before identifying the relevant rule or given information
  • Dropping a sign, unit, restriction, or exact-value symbol during simplification
  • Comparing only the final answer instead of checking the first incorrect step
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