Grade 12 · Math · Calculus

12th grade Integration by substitution worksheet

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Integration by substitution

Grade 12 Math Worksheet - Challenge Me

NameDate

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

Jun writes ∫ x(4x² + 5)² dx = (4x² + 5)³/N + C. What is N?
Kira writes ∫ x(5x² + 2)³ dx = (5x² + 2)⁴/N + C. What is N?
Luz writes ∫ x(2x² + 7)⁴ dx = (2x² + 7)⁵/N + C. What is N?
Milo evaluates ∫ from 0 to π/2 of cos x sin² x dx with u = sin x. What is it?
Noah evaluates ∫ from 0 to π/2 of cos x sin³ x dx with u = sin x. What is it?
Zee evaluates ∫ from 0 to π/2 of cos x sin⁵ x dx with u = sin x. What is it?
7.
Which substitution turns ∫ 4x³(x⁴ − 2)⁶ dx into a basic integral in u, for Tala?
- u = 4x³, du = 12x² dx
- u = x, du = dx
- u = x⁴ − 2, du = 4x³ dx
8.
Which substitution turns ∫ e^(√x)/√x dx into a basic integral in u, for Pia?
- u = √x, du = dx/(2√x)
- u = 1/√x, du = −dx/(2x√x)
- u = x, du = dx
9.
Which substitution turns ∫ x/(x² + 9) dx into a basic integral in u, for Esme?
- u = x, du = dx
- u = 2x, du = 2 dx, so x = u/2
- u = x² + 9, du = 2x dx
10.
Which substitution turns ∫ (ln x)²/x dx into a basic integral in u, for Jun?
- u = 1/x, du = −dx/x²
- u = x, du = dx
- u = ln x, du = dx/x

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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