10 questions · Form 5 Additional Mathematics Bab 3: Integration
Find the indefinite integral ∫ (3x² - 4x + 5) dx.
Prefer reading to quizzing? All 10 questions are listed below with the answer and explanation under each one.
1. Find the indefinite integral ∫ (3x² - 4x + 5) dx.
Answer: A
∫ (3x² - 4x + 5) dx = 3(x³/3) - 4(x²/2) + 5x + c = x³ - 2x² + 5x + c.
2. Evaluate ∫₀⁴ (1 / √(2x + 1)) dx.
Answer: A
∫₀⁴ (2x + 1)⁻¹/² dx = [(2x + 1)¹/² / (2 × 12)]₀⁴ = [√(2x + 1)]₀⁴ = √9 - √1 = 3 - 1 = 2.
3. Evaluate ∫ (2x + 1)³ dx.
Answer: B
Using ∫ (ax + b)ⁿ dx = (ax + b)ⁿ⁺¹ / [a(n + 1)] + c: ∫ (2x + 1)³ dx = (2x + 1)⁴ / [2(4)] + c = (2x + 1)⁴ / 8 + c.
4. Find the area bounded by the curve x = y² - 4 and the y-axis from y = 0 to y = 2.
Answer: A
Area = |∫₀² (y² - 4) dy| = |[y³/3 - 4y]₀²| = |(83 - 8) - 0| = |-163| = 163 units².
5. Find the volume of revolution generated when the region bounded by the curve y = √x, the x-axis, and x = 4 is rotated 360° about the x-axis.
Answer: A
V = π ∫₀⁴ y² dx = π ∫₀⁴ (√x)² dx = π ∫₀⁴ x dx = π [x²/2]₀⁴ = π (162 - 0) = 8π units³.
6. Given that ∫₁⁴ f(x) dx = 7, evaluate ∫₁⁴ [2f(x) + 3] dx.
Answer: A
∫₁⁴ [2f(x) + 3] dx = 2 ∫₁⁴ f(x) dx + ∫₁⁴ 3 dx = 2(7) + [3x]₁⁴ = 14 + (12 - 3) = 14 + 9 = 23.
7. Given that ∫₂⁵ g(x) dx = 6 and ∫₅⁸ g(x) dx = -2, find the value of ∫₂⁸ g(x) dx.
Answer: C
Using property: ∫₂⁸ g(x) dx = ∫₂⁵ g(x) dx + ∫₅⁸ g(x) dx = 6 + (-2) = 4.
8. Find the volume generated when the line y = 2x from x = 0 to x = 3 is revolved 360° about the x-axis.
Answer: A
V = π ∫₀³ y² dx = π ∫₀³ (2x)² dx = π ∫₀³ 4x² dx = π [4x³/3]₀³ = π [4273] = 36π units³.
9. Find ∫ (6x³ - 2) dx.
Answer: A
∫ (6x⁻³ - 2) dx = 6(x⁻²/-2) - 2x + c = -3x⁻² - 2x + c = -3x² - 2x + c.
10. If d/dx [f(x)] = 3x² + 2x, find f(x) given that f(1) = 4.
Answer: A
f(x) = ∫ (3x² + 2x) dx = x³ + x² + c. f(1) = 1³ + 1² + c = 4 => 2 + c = 4 => c = 2. f(x) = x³ + x² + 2.