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: d/dx(x³ + x² - x + C) = 3x² + 2x - 1 ✓ CONCLUSION: ∫ (3x² + 2x - 1) dx = x³ + x² - x + C

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 origin. The area under this line from x = 0 to x = 1 forms a triangle with base 1 and height 2. Area = (1/2) × 1 × 2 = 1 ✓ CONCLUSION: ∫₀¹ 2x dx = 1

We need to find the antiderivative of the exponential function e^x. The exponential function has the special property that it is its own derivative: de^x/dx = e^x Therefore, the antiderivative is: ∫ e^x dx = e^x + C VERIFICATION: d/dx(e^x + C) = e^x ✓ CONCLUSION: ∫ e^x dx = e^x + C This is a fundamental formula in calculus.

We need to evaluate the definite integral of sin(x) from 0 to π/2. Step 1: Find the antiderivative We know that d/dx(-cos(x)) = sin(x) So ∫ sin(x) dx = -cos(x) + C Step 2: Apply the Fundamental Theorem of Calculus ∫₀^(π/2) sin(x) dx = [-cos(x)]₀^(π/2) = (-cos(π/2)) - (-cos(0)) = (-0) - (-1) = 0 + 1 = 1 GEOMETRIC INTERPRETATION: The integral represents the area under the sine curve from x = 0 to x = π/2, which is indee...

We need to find the antiderivative of 1/x. Note: This is a special case where the power rule doesn't apply directly because n = -1. We use the logarithmic integration rule: ∫ (1/x) dx = ln|x| + C For x > 0, the absolute value can be dropped: ∫ (1/x) dx = ln(x) + C VERIFICATION: d/dx(ln(x)) = 1/x ✓ CONCLUSION: ∫ (1/x) dx = ln(x) + C (for x > 0) ∫ (1/x) dx = ln|x| + C (for x ≠ 0, general form)