Integration

Find ∫ (6x² + 4x) dx.

• A2x³ + 4x² + c
• B6x³ + 2x² + c
• C12x + 4 + c
• D2x³ + 2x² + c✓

Answer: Integrating term by term: ∫6x² dx = 2x³ and ∫4x dx = 2x², giving 2x³ + 2x² + c.

Evaluate the definite integral ∫₁³ 2x dx.

• A6
• B4
• C9
• D8✓

Answer: ∫2x dx = x², so evaluating from 1 to 3 gives 3² − 1² = 9 − 1 = 8.

Find the area between the curve y = 3x² and the x-axis from x = 0 to x = 2.

• A6
• B12
• C24
• D8✓

Answer: Area = ∫₀² 3x² dx = [x³]₀² = 8 − 0 = 8.

Evaluate ∫₀⁴ (2x + 1) dx.

• A24
• B18
• C20✓
• D16

Answer: ∫(2x + 1) dx = x² + x, so from 0 to 4 the value is (16 + 4) − 0 = 20.

Find ∫ (8x³ − 6x + 5) dx.

• A24x² − 6 + c
• B2x⁴ − 6x² + 5x + c
• C2x⁴ − 3x² + 5x
• D2x⁴ − 3x² + 5x + c✓

Answer: Integrate term by term: ∫8x³ dx = 2x⁴, ∫−6x dx = −3x², ∫5 dx = 5x, plus the constant. So the answer is 2x⁴ − 3x² + 5x + c. The 24x² − 6 + c distractor differentiates instead; 2x⁴ − 6x² + 5x + c mishandles the −6x term; 2x⁴ − 3x² + 5x omits +c.

Find ∫ (4/x² + 3√x) dx. (Write 4/x² as 4x^(−2) and √x as x^(1/2).)

• A−8x^(−3) + (3/2)x^(−1/2) + c
• B4x^(−1) + 2x^(3/2) + c
• C−4x^(−1) + (3/2)x^(3/2) + c
• D−4x^(−1) + 2x^(3/2) + c✓

Answer: ∫4x^(−2) dx = 4x^(−1)/(−1) = −4x^(−1) = −4/x. ∫3x^(1/2) dx = 3x^(3/2)/(3/2) = 2x^(3/2). So the answer is −4/x + 2x^(3/2) + c. The −8x^(−3) + (3/2)x^(−1/2) + c distractor differentiates; 4x^(−1) + 2x^(3/2) + c has a sign error; −4x^(−1) + (3/2)x^(3/2) + c mishandles the coefficient of x^(3/2).

Evaluate the definite integral ∫₁² (3x² + 2) dx.

• A9✓
• B7
• C11
• D8

Answer: ∫(3x² + 2) dx = x³ + 2x. Evaluating: at x = 2, 8 + 4 = 12; at x = 1, 1 + 2 = 3. So the integral is 12 − 3 = 9. B forgets the +2 term contribution; the others come from arithmetic slips.

Find the area enclosed between the curve y = x² and the line y = 2x.

• A8/3
• B4/3✓
• C2/3
• D16/3

Answer: They meet where x² = 2x, i.e. x = 0 and x = 2. Between these, 2x ≥ x², so area = ∫₀² (2x − x²) dx = [x² − x³/3]₀² = (4 − 8/3) = 12/3 − 8/3 = 4/3. A integrates the wrong difference; C and D come from limit/sign errors.

Use the substitution u = x² + 1 to find ∫ 2x(x² + 1)³ dx.

• A(x² + 1)⁴ + c
• B(1/4)(x² + 1)⁴ + c✓
• C(1/2)(x² + 1)⁴ + c
• D8x²(x² + 1)² + c

Answer: With u = x² + 1, du = 2x dx, the integral becomes ∫u³ du = u⁴/4 + c = (1/4)(x² + 1)⁴ + c. A forgets to divide by 4; C divides by 2 instead of 4; D differentiates.

Use integration by parts to find ∫ x e^x dx.

• Ax e^x + e^x + c
• B(x²/2) e^x + c
• Cx e^x − e^x + c✓
• Dx e^x + c

Answer: Take u = x (du = dx) and dv = e^x dx (v = e^x). Then ∫x e^x dx = x e^x − ∫e^x dx = x e^x − e^x + c. The x e^x + e^x + c distractor has the wrong sign on the second term; (x²/2) e^x + c integrates x as if e^x were constant; x e^x + c omits the second term entirely.

A curve passes through the point (2, 5) and has gradient function dy/dx = 3x² − 4x. Find the equation of the curve.

• Ay = x³ − 2x² + 1
• By = x³ − 2x² + 5✓
• Cy = 6x − 4 + 5
• Dy = x³ − 2x² + 9

Answer: Integrate: y = x³ − 2x² + c. Substitute (2, 5): 5 = 8 − 8 + c, so c = 5. Thus y = x³ − 2x² + 5. The c = 1 version uses the wrong constant; the c = 9 version uses the wrong constant; y = 6x − 4 + 5 differentiates the gradient function instead of integrating.

Find ∫ (2x + 3)⁴ dx using the reverse chain rule.

• A(1/10)(2x + 3)⁵ + c✓
• B(1/5)(2x + 3)⁵ + c
• C(2x + 3)⁵ + c
• D8(2x + 3)³ + c

Answer: Raise the power and divide by the new power and the inner coefficient: ∫(2x + 3)⁴ dx = (2x + 3)⁵/(5 × 2) + c = (1/10)(2x + 3)⁵ + c. B forgets to divide by the inner coefficient 2; C divides by neither; D differentiates.