Evaluate ∫10 4x – 6 3√x2 dx

JAMB Mathematics 2023 45 views

Evaluate ∫₀¹ (4x – 6∛(x²)) dx

  1. ⁸⁄₅
  2. -⁵⁄₈
  3. -⁸⁄₅ ✓
  4. ⁵⁄₈

Explanation

Step 1: Rewrite using exponent notation
∛(x²) = x^(2/3)
Integral becomes: ∫₀¹ (4x – 6x^(2/3)) dx

Step 2: Integrate first term
∫4x dx = 4 × x²/2 = 2x²

Step 3: Integrate second term
∫6x^(2/3) dx = 6 × x^(2/3 + 1)/(2/3 + 1)
= 6 × x^(5/3)/(5/3)
= 6 × (3/5) × x^(5/3)
= (18/5)x^(5/3)

Step 4: Combine and apply limits
[2x² – (18/5)x^(5/3)]₀¹
At x = 1: 2(1)² – (18/5)(1)^(5/3) = 2 – 18/5
= 10/5 – 18/5 = -8/5
At x = 0: Everything equals 0

Step 5: Final answer
-8/5 – 0 = -8/5

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