3-2 Methods of integration: practice questions

Correct answer: (C) 2x⁵ + c

Explanation: Step 1: ∫10x⁴ dx = 10·(x⁵/5) + c. Step 2: = 2x⁵ + c. ∴ ∫10x⁴ dx = 2x⁵ + c

Correct answer: (A) 4x + c

Explanation: Step 1: The integral of a constant k is kx. Step 2: ∫4 dx = 4x + c; check: d/dx(4x) = 4. ∴ ∫4 dx = 4x + c

Correct answer: (B) x³ + c

Explanation: Step 1: ∫3x² dx = 3∫x² dx = 3·(x³/3) + c. Step 2: = x³ + c. ∴ ∫3x² dx = x³ + c

Correct answer: (A) −(1/2)cos(2x) + c

Explanation: Step 1: Use ∫sin(ax) dx = −(1/a)cos(ax) + c, with a = 2. Step 2: = −(1/2)cos(2x) + c. ∴ ∫sin(2x) dx = −(1/2)cos(2x) + c

Correct answer: (B) 2e^(x/2)(x − 2) + c

Explanation: Step 1: Let y = x, dz = e^(x/2) dx, so dy = dx and z = 2e^(x/2). Step 2: ∫ y dz = yz − ∫ z dy = 2x e^(x/2) − ∫ 2e^(x/2) dx. Step 3: = 2x e^(x/2) − 4e^(x/2) + c. Step 4: ∴ 2e^(x/2)(x − 2) + c

Correct answer: (A) (x²+1)⁴/8 + c

Explanation: Step 1: Let u = x²+1 ⇒ du = 2x dx ⇒ x dx = du/2. Step 2: ∫u³·(du/2) = (1/2)·u⁴/4 = u⁴/8. Step 3: Back-substitute: = (x²+1)⁴/8 + c. ∴ ∫x(x²+1)³ dx = (x²+1)⁴/8 + c

Correct answer: (A) (x²+5)⁵/5 + c

Explanation: Step 1: Let u = x²+5 ⇒ du = 2x dx. Step 2: ∫2x(x²+5)⁴ dx = ∫u⁴ du = u⁵/5. Step 3: = (x²+5)⁵/5 + c. ∴ ∫2x(x²+5)⁴ dx = (x²+5)⁵/5 + c

Correct answer: (D) 3

Explanation: Step 1: The curve is f(x) = ∫ x eˣ dx. By parts (differentiate x, integrate eˣ dx): f(x) = x eˣ − ∫ eˣ dx = eˣ(x − 1) + c. Step 2: At (0, 2): 2 = e⁰(0 − 1) + c ⇒ c = 3, so f(x) = eˣ(x − 1) + 3. Step 3: f(1) = e(0) + 3 = 3. ∴ 3 (0 forgets the constant c; e + 2 integrates x eˣ as x eˣ only).

Correct answer: (A) (2/9)(x³+1)√(x³+1) + c

Explanation: Step 1: Let u = x³+1 ⇒ du = 3x² dx ⇒ x² dx = du/3. Step 2: ∫√u·(du/3) = (1/3)·u^(3/2)/(3/2) = (2/9)u^(3/2). Step 3: = (2/9)(x³+1)√(x³+1) + c. ∴ ∫x²√(x³+1) dx = (2/9)(x³+1)√(x³+1) + c

Correct answer: (A) (1/2)(ln x)² + c

Explanation: Step 1: Let u = ln x ⇒ du = (1/x) dx. Step 2: ∫u du = u²/2. Step 3: = (1/2)(ln x)² + c. ∴ ∫(ln x)/x dx = (1/2)(ln x)² + c