Integrals — Class 12 Mathematics NCERT Solutions (Free)

We need to find the antiderivative of 5x⁴. Using the power rule for integration: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (where n ≠ -1) ∫ 5x⁴ dx = 5 ∫ x⁴ dx = 5 × (x⁴⁺¹/(4+1)) + C = 5 × (x⁵/5) + C = x⁵ + C VERIFICATION: d/dx(x⁵ + C) = 5x⁴ ✓ CONCLUSION: ∫ 5x⁴ dx = x⁵ + C

We integrate each term separately using the power rule. ∫ (3x² + 2x - 1) dx = ∫ 3x² dx + ∫ 2x dx - ∫ 1 dx Term 1: ∫ 3x² dx = 3 × (x²⁺¹/(2+1)) = 3 × (x³/3) = x³ Term 2: ∫ 2x dx = 2 × (x¹⁺¹/(1+1)) = 2 × (x²/2) = x² Term 3: ∫ 1 dx = x Combining: ∫ (3x² + 2x - 1) dx = x³ + x² - x + C VERIFICATION:...

We need to evaluate the definite integral from 0 to 1 of 2x. Step 1: Find the antiderivative ∫ 2x dx = 2 × (x²/2) = x² + C Step 2: Apply the Fundamental Theorem of Calculus ∫₀¹ 2x dx = [x²]₀¹ = (1)² - (0)² = 1 - 0 = 1 GEOMETRIC INTERPRETATION: The function y = 2x is a straight line through the or...