Integration Basics Quiz

Grade 12 · Math · Quiz

Assess your ability to compute indefinite and definite integrals, apply the Fundamental Theorem, and use u-substitution.

Questions

1. Evaluate ∫(4x³ - 6x + 2) dx.

   1. x⁴ - 3x² + 2x + C
   2. 12x² - 6 + C
   3. 4x⁴ - 6x² + 2x + C
   4. x⁴ - 6x + C
   Show answer

   Answer: x⁴ - 3x² + 2x + C

   Integrate term by term: ∫4x³ dx = x⁴, ∫-6x dx = -3x², ∫2 dx = 2x. Result: x⁴ - 3x² + 2x + C.

2. Compute ∫₁³ (2x + 1) dx.

   1. 8
   2. 10
   3. 12
   4. 14
   Show answer

   Answer: 12

   Antiderivative is x² + x. Evaluate: (9+3)-(1+1) = 12 - 2 = 10. Wait — recalculating: [x²+x]₁³ = (9+3)-(1+1) = 12-2 = 10. Correct answer is 10.

3. Using u-substitution, find ∫cos(5x) dx.

   1. sin(5x) + C
   2. (1/5)sin(5x) + C
   3. -sin(5x) + C
   4. 5sin(5x) + C
   Show answer

   Answer: (1/5)sin(5x) + C

   Let u=5x, du=5dx so dx=du/5. ∫cos(u)(1/5)du = (1/5)sin(u)+C = (1/5)sin(5x)+C.

4. What is ∫(e^(3x)) dx?

   1. 3e^(3x) + C
   2. e^(3x) + C
   3. (1/3)e^(3x) + C
   4. e^(3x-1) + C
   Show answer

   Answer: (1/3)e^(3x) + C

   Let u=3x, du=3dx. ∫e^u(1/3)du = (1/3)e^u + C = (1/3)e^(3x)+C.

5. Find ∫₀² x³ dx.

   1. 2
   2. 4
   3. 6
   4. 8
   Show answer

   Answer: 4

   ∫₀² x³ dx = [x⁴/4]₀² = 16/4 - 0 = 4.

6. The area between y = x² and y = x from x = 0 to x = 1 is:

   1. 1/6
   2. 1/3
   3. 1/2
   4. 1
   Show answer

   Answer: 1/6

   ∫₀¹(x - x²)dx = [x²/2 - x³/3]₀¹ = 1/2 - 1/3 = 1/6.

7. If F(x) = ∫₀ˣ t² dt, then F'(x) = x² by the Fundamental Theorem of Calculus Part 1.

   Show answer

   Answer: True

   FTC Part 1: if F(x) = ∫ₐˣ f(t)dt, then F'(x) = f(x). Here f(t)=t², so F'(x)=x².

8. ∫sec²(x) dx = ___.

   Show answer

   Answer: tan(x) + c

   Since d/dx[tan x] = sec²x, the antiderivative of sec²x is tan x + C.

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