Grade 12 · Math · Calculus

12th grade Integration by substitution worksheet

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Integration by substitution

Grade 12 Math Worksheet

NameDate

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

1.
Esme uses a substitution. Which is ∫ 2x cos(x²) dx?
- 2x sin(x²) + C
- (1/2)sin(x²) + C
- sin(x²) + C
2.
Jun uses a substitution. Which is ∫ x e^(x²) dx?
- e^(x²) + C
- (1/2)e^(x²) + C
- (1/2)x e^(x²) + C
3.
Kira uses a substitution. Which is ∫ cos x sin⁴ x dx?
- (1/5)sin⁵ x + C
- sin⁵ x + C
- cos x · (1/5)sin⁵ x + C
4.
Luz uses a substitution. Which is ∫ x/(x² + 1) dx?
- ln(x² + 1) + C
- (1/2)ln(x² + 1) + C
- (1/2)x ln(x² + 1) + C
Noah evaluates ∫ from 0 to 2 of 2x(x² + 6)² dx with u = x² + 6. What is it?
Ren evaluates ∫ from 0 to 1 of 2x(x² + 2)² dx with u = x² + 2. What is it?
Zee evaluates ∫ from 0 to 2 of 2x(x² + 3)² dx with u = x² + 3. What is it?
8.
Tala substitutes the inside part as u in ∫ from 0 to 2 of 2x(x² + 1)³ dx. Which integral in u is right?
- ∫ from 1 to 5 of 2xu³ du
- ∫ from 1 to 5 of u³ du
- ∫ from 0 to 2 of u³ du
9.
Esme substitutes the inside part as u in ∫ from 0 to π/2 of cos x sin² x dx. Which integral in u is right?
- ∫ from 0 to 1 of u² du
- ∫ from 0 to π/2 of u² du
- ∫ from 0 to 1 of u² cos x du
10.
Jun substitutes the inside part as u in ∫ from 1 to 2 of 2x e^(x²) dx. Which integral in u is right?
- ∫ from 1 to 4 of 2x eᵘ du
- ∫ from 1 to 4 of eᵘ du
- ∫ from 1 to 2 of eᵘ du

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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