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U-Substitution - Worksheet 9 - 12th Grade

Hard

Compute exact integrals and sums, distinguish area from signed integrals, and include constants for indefinite integrals.

hard: Use another set to practice explaining and checking your answers.

27 problems · 25-30 minutes · Skills: integrals, u-substitution, calculus

About This Worksheet

This hard-level worksheet contains 27 problems designed for 12th Grade students practicing integration by substitution (u-sub). Hard worksheets feature challenging problems including multi-step questions, missing values, and real-world applications to push students further.

Manipulate algebraic expressions and understand functions, limits and derivative notation.

  • Compute exact integrals and sums, distinguish area from signed integrals, and include constants for indefinite integrals.
  • Check your result using the method in the example before moving to the next question.

How to Solve: Worked Example

Example Problem: Evaluate ∫ 6x(3x² + 2)^3 dx.
Solution Steps:
  1. Let u = 3x² + 2, so du = 6x dx.
  2. Integrate u raised to the given power, then substitute back.
  3. Add C.
Answer: (3x² + 2)^4/4 + C
Tip: Check your result using the method in the example before moving to the next question.
1
Evaluate ∫ 4x(2x² + 5)^1 dx.
2
Evaluate ∫ 2x(1x² + 2)^2 dx.
3
Evaluate ∫ 4x(2x² + 1)^4 dx.
4
Evaluate ∫ 2x(1x² + 5)^4 dx.
5
Evaluate ∫ 6x(3x² + 1)^2 dx.
6
Evaluate ∫ 10x(5x² + 3)^1 dx.
7
Evaluate ∫ 2x(1x² + 4)^3 dx.
8
Evaluate ∫ 8x(4x² + 5)^2 dx.
9
Evaluate ∫ 2x(1x² + 2)^1 dx.
10
Evaluate ∫ 8x(4x² + 4)^3 dx.
11
Evaluate ∫ 8x(4x² + 1)^2 dx.
12
Evaluate ∫ 12x(6x² + 5)^3 dx.
13
Evaluate ∫ 4x(2x² + 1)^3 dx.
14
Evaluate ∫ 12x(6x² + 1)^4 dx.
15
Evaluate ∫ 8x(4x² + 3)^4 dx.
16
Evaluate ∫ 2x(1x² + 1)^3 dx.
17
Evaluate ∫ 12x(6x² + 3)^2 dx.
18
Evaluate ∫ 4x(2x² + 1)^2 dx.
19
Evaluate ∫ 8x(4x² + 2)^4 dx.
20
Evaluate ∫ 12x(6x² + 1)^2 dx.
21
Evaluate ∫ 10x(5x² + 1)^4 dx.
22
Evaluate ∫ 4x(2x² + 3)^4 dx.
23
Evaluate ∫ 6x(3x² + 5)^2 dx.
24
Evaluate ∫ 2x(1x² + 5)^1 dx.
25
Evaluate ∫ 12x(6x² + 4)^2 dx.
26
Evaluate ∫ 8x(4x² + 1)^1 dx.
27
Evaluate ∫ 4x(2x² + 4)^3 dx.

Answer Key

1. (2x² + 5)^2/2 + C2. (1x² + 2)^3/3 + C3. (2x² + 1)^5/5 + C4. (1x² + 5)^5/5 + C5. (3x² + 1)^3/3 + C6. (5x² + 3)^2/2 + C7. (1x² + 4)^4/4 + C8. (4x² + 5)^3/3 + C9. (1x² + 2)^2/2 + C10. (4x² + 4)^4/4 + C11. (4x² + 1)^3/3 + C12. (6x² + 5)^4/4 + C13. (2x² + 1)^4/4 + C14. (6x² + 1)^5/5 + C15. (4x² + 3)^5/5 + C16. (1x² + 1)^4/4 + C17. (6x² + 3)^3/3 + C18. (2x² + 1)^3/3 + C19. (4x² + 2)^5/5 + C20. (6x² + 1)^3/3 + C21. (5x² + 1)^5/5 + C22. (2x² + 3)^5/5 + C23. (3x² + 5)^3/3 + C24. (1x² + 5)^2/2 + C25. (6x² + 4)^3/3 + C26. (4x² + 1)^2/2 + C27. (2x² + 4)^4/4 + C

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What skills does this integrals worksheet cover?

This worksheet focuses on integrals, u-substitution, calculus. It is designed for 12th Grade students and provides structured practice through varied problem types with a worked example to guide students before they begin.

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How many problems are on this worksheet?

This worksheet contains 27 problems. It typically takes 25-30 minutes to complete. An answer key with all solutions is included so parents and teachers can check work quickly.