Integration by substitution

Answer Key - Grade 12 Math Worksheet

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

1. 1.

   Esme uses a substitution. Which is ∫ 2x cos(x²) dx?

   • 2x sin(x²) + C
   • (1/2)sin(x²) + C
   • sin(x²) + C
2. 2.

   Jun uses a substitution. Which is ∫ x e^(x²) dx?

   • (1/2)e^(x²) + C
   • (1/2)x e^(x²) + C
   • e^(x²) + C
3. 3.

   Kira uses a substitution. Which is ∫ cos x sin⁴ x dx?

   • sin⁵ x + C
   • (1/5)sin⁵ x + C
   • cos x · (1/5)sin⁵ x + C
4. 4.

   Luz uses a substitution. Which is ∫ x/(x² + 1) dx?

   • (1/2)x ln(x² + 1) + C
   • ln(x² + 1) + C
   • (1/2)ln(x² + 1) + C
5. 5.

   Milo evaluates ∫ from 0 to 1 of 2x(x² + 1)² dx with u = x² + 1. What is it?

   7/3
6. 6.

   Noah evaluates ∫ from 0 to 2 of 2x(x² + 6)² dx with u = x² + 6. What is it?

   784/3
7. 7.

   Zee evaluates ∫ from 0 to 2 of 2x(x² + 3)² dx with u = x² + 3. What is it?

   316/3
8. 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. 9.

   Pia substitutes the inside part as u in ∫ from 0 to 1 of 3x²(x³ + 2)⁴ dx. Which integral in u is right?

   • ∫ from 2 to 3 of 3x²u⁴ du
   • ∫ from 0 to 1 of u⁴ du
   • ∫ from 2 to 3 of u⁴ du
10. 10.

    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 π/2 of u² du
    • ∫ from 0 to 1 of u² cos x du
    • ∫ from 0 to 1 of u² du