Integration Basics Quiz

Grade 12 · Math · Quiz

10 questions on antiderivatives, definite integrals, and u-substitution.

Questions

1. What is ∫ 6x² dx?

   1. 2x³ + C
   2. 3x³ + C
   3. 6x³ + C
   4. 2x² + C
   Show answer

   Answer: 2x³ + C

   ∫ 6x² dx = 6 · x³/3 + C = 2x³ + C.

2. Evaluate ∫₁³ (2x + 1) dx.

   Show answer

   Antiderivative: x² + x. FTC: [x²+x]₁³ = (9+3) - (1+1) = 12 - 2 = 10.

3. What is ∫ cos x dx?

   1. sin x + C
   2. −sin x + C
   3. −cos x + C
   4. sec x + C
   Show answer

   Answer: sin x + C

   d/dx[sin x] = cos x, so ∫ cos x dx = sin x + C.

4. ∫ₐᵇ f(x) dx = −∫ᵦᵃ f(x) dx.

   Show answer

   Reversing the limits of integration negates the integral: F(a)-F(b) = −(F(b)-F(a)).

5. What is ∫ 1/x dx?

   1. x⁻² + C
   2. ln|x| + C
   3. 1/x² + C
   4. x + C
   Show answer

   Answer: ln|x| + C

   The power rule doesn't apply when n = −1. Instead, ∫ 1/x dx = ln|x| + C.

6. Using u = 2x+3, evaluate ∫ 2(2x+3)³ dx.

   Show answer

   du = 2 dx, so ∫ u³ du = u⁴/4 + C = (2x+3)⁴/4 + C.

7. Evaluate ∫₀^π sin x dx.

   1. 0
   2. 1
   3. 2
   4. π
   Show answer

   Answer: 2

   [−cos x]₀^π = −cos π − (−cos 0) = −(−1) + 1 = 2.

8. The constant of integration C is necessary in definite integrals.

   Show answer

   In a definite integral, C cancels out: F(b)+C − (F(a)+C) = F(b) − F(a). So C is not needed.

9. By FTC Part 1, d/dx[∫₀ˣ √(t²+1) dt] = ?

   1. √(x²+1)
   2. x√(x²+1)
   3. √(t²+1)
   4. 0
   Show answer

   Answer: √(x²+1)

   FTC Part 1: differentiating an integral with x as upper limit returns the integrand evaluated at x.

10. What is ∫ eˣ + x dx?

    Show answer

    Integrate term by term: ∫ eˣ dx = eˣ, ∫ x dx = x²/2. Total: eˣ + x²/2 + C.

Get a personalized course on Integration Basics built around your child.