Grade 12 · Math · Calculus

12th grade Integration by substitution worksheet

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Integration by substitution

Grade 12 Math Worksheet - Building Confidence

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Circle the best answer, or write your answer on the line.

1.
Esme wants a substitution for ∫ 2x(x² + 1)⁵ dx. Which u and du make it a basic integral?
- u = 2x, du = 2 dx
- u = x² + 1, du = 2x dx
- u = x² + 1, du = 2x² dx
2.
Jun wants a substitution for ∫ 3x²√(x³ + 4) dx. Which u and du make it a basic integral?
- u = x³ + 4, du = 3x³ dx
- u = x³ + 4, du = 3x² dx
- u = 3x², du = 6x dx
3.
Kira wants a substitution for ∫ (2x + 3)(x² + 3x)⁴ dx. Which u and du make it a basic integral?
- u = x² + 3x, du = (2x + 3x) dx
- u = x² + 3x, du = (2x + 3) dx
- u = 2x + 3, du = 2 dx
4.
Luz wants a substitution for ∫ cos x · e^(sin x) dx. Which u and du make it a basic integral?
- u = cos x, du = −sin x dx
- u = sin x, du = cos x dx
- u = sin x, du = −cos x dx
Noah writes ∫ x(x² + 5)⁴ dx = a(x² + 5)⁵ + C. What is a?
Ren writes ∫ x(x² + 2)⁵ dx = a(x² + 2)⁶ + C. What is a?
Zee writes ∫ x(x² + 7)² dx = a(x² + 7)³ + C. What is a?
Tala substitutes u = x² + 1 in ∫ from 0 to 2 of 2x(x² + 1)³ dx. What is the new upper limit for u?
Pia substitutes u = x² + 4 in ∫ from 0 to 3 of 2x(x² + 4)³ dx. What is the new upper limit for u?
Esme substitutes u = x² + 2 in ∫ from 0 to 4 of 2x(x² + 2)³ dx. What is the new upper limit for u?

Integration by substitution, indefinite and definite. Every version has 10 questions, and the answer key prints on the second page.

Standard: none (beyond CCSS)

substitution: Renaming the inside part u to make an integral basic.

du: u′ dx, the small change in u.

new limits: The u-values at the old x-limits.

Chose the factor outside as u. u should be the inside part whose derivative sits outside.

Picked the right u but differentiated it wrongly, so du does not match the integral.

Did not divide by 2. With u = x² + c, x dx is du/2.

What is the most common mistake with integration by substitution?

Chose the factor outside as u. u should be the inside part whose derivative sits outside.

Are these 12th grade math worksheets free to print?

Yes. All 6 printable versions at each level are free to print, with the answer key, and no account is needed.

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