Pure Mathematics — Second: Differential & Integral Calculus · Mathematics. These are 8 questions from the Magmo3 bank on this lesson, from easy to hard. Solve each one, then open "Show answer and explanation". Magmo3 has 38 questions on this lesson, each with an explanation.
Given ∫₀⁴ f(x) dx = 10 and ∫₀⁴ g(x) dx = 3, find ∫₀⁴ [f(x) − g(x)] dx
Correct answer: (A) 7
Explanation: Step 1: (property) the integral of a difference equals the difference of the integrals. Step 2: ∫₀⁴ [f − g] dx = ∫₀⁴ f dx − ∫₀⁴ g dx. Step 3: = 10 − 3. ∴ the value = 7
Evaluate ∫₋₁¹ x³ dx
Correct answer: (B) 0
Explanation: Step 1: (property) x³ is an odd function, so its integral over a symmetric interval [−1, 1] is 0. Step 2: check: ∫ x³ dx = x⁴/4, [x⁴/4] from −1 to 1. Step 3: (1/4) − (1/4) = 0. ∴ the value = 0
Evaluate the definite integral of sin x from x = 0 to x = π/2
Correct answer: (B) 1
Explanation: Step 1: (set up) ∫ sin x dx = −cos x. Step 2: [−cos x] from 0 to π/2. Step 3: (−cos π/2) − (−cos 0) = 0 − (−1). ∴ the value = 1
∫₀^(ln 2) e^(2x) dx = ..........
Correct answer: (C) 3/2
Explanation: Step 1: ∫ e^(2x) dx = (1/2)e^(2x). Step 2: [(1/2)e^(2x)] from 0 to ln 2 = (1/2)(e^(2 ln 2) − e⁰) = (1/2)(4 − 1). Step 3: ∴ 3/2
If ∫₂⁷ f(x) dx = 9, find ∫₇² f(x) dx
Correct answer: (D) −9
Explanation: Step 1: (property) reversing the limits changes the sign: ∫₇² f dx = −∫₂⁷ f dx. Step 2: substitute the given value: = −(9). Step 3: simplify. ∴ the value = −9
∫₀³ x √(x² + 16) dx = ..........
Correct answer: (B) 61/3
Explanation: Step 1: Put z = x² + 16, dz = 2x dx; when x = 0, z = 16 and when x = 3, z = 25. Step 2: The integral = (1/2) ∫₁₆²⁵ z^(1/2) dz = (1/2) × (2/3) [z^(3/2)] from 16 to 25. Step 3: = (1/3)(125 − 64) = 61/3. Step 4: ∴ 61/3
Evaluate ∫₀¹ (4x³ − 3x² + 2x) dx
Correct answer: (D) 1
Explanation: Step 1: (set up) ∫ (4x³ − 3x² + 2x) dx = x⁴ − x³ + x². Step 2: [x⁴ − x³ + x²] from 0 to 1. Step 3: (1 − 1 + 1) − 0 = 1. ∴ the value = 1
If ∫₋ₖᵏ (3x² + 2x⁵ + 1) dx = 20, where k is greater than 0, then k = ..........
Correct answer: (D) 2
Explanation: Step 1: 2x⁵ is an odd function, so ∫₋ₖᵏ 2x⁵ dx = 0. Step 2: 3x² + 1 is an even function, so ∫₋ₖᵏ (3x² + 1) dx = 2 ∫₀ᵏ (3x² + 1) dx = 2(k³ + k). Step 3: 2(k³ + k) = 20, so k³ + k − 10 = 0, i.e. (k − 2)(k² + 2k + 5) = 0. Step 4: k² + 2k + 5 has no real roots. Step 5: ∴ k = 2
The first 10 questions in every subject are free, each with an explanation.