INTEGRATION O LEVEL

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๐Ÿ“ INTEGRATION - Theory & Formulas

O Level Mathematics | Complete Guide

1. Definition

Integration = Reverse of Differentiation

DIFFERENTIATION ↓ f(x) = x³ → f'(x) = 3x² ↑ INTEGRATION
⚠️ Important: Integration is many-to-one process. Different functions have same derivative!
x² + 5 → 2x | x² + 10 → 2x | x² - 3 → 2x

2. Three Basic Rules

Rule 1: Power Rule ⭐

∫ xn dx = xn+1/(n+1) + c
Steps: (1) Add 1 to power (2) Divide by new power (3) Add + c
Ex: ∫ x³ dx = x⁴/4 + c | ∫ x⁷ dx = x⁸/8 + c | ∫ 10x⁴ dx = 2x⁵ + c

Rule 2: Constant Multiplier

∫ k·f(x) dx = k ∫ f(x) dx

Rule 3: Sum/Difference

∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx
Ex: ∫ (x³ - x + 3) dx = x⁴/4 - x²/2 + 3x + c

3. Constant of Integration (+c)

Why +c? Because many functions have same derivative! Always add +c for indefinite integrals. For definite integrals (with limits), NO +c needed.
  • Indefinite: ∫ f(x) dx = F(x) + c ✓
  • Twice: Use c₁ and c₂
  • Definite: ∫โ‚แต‡ f(x) dx = [F(x)]โ‚แต‡ (no c)

4. Function Types

A. Negative Powers (Fractions)

1/xn = x-n
Ex: ∫ 8/x⁵ dx = ∫ 8x-5 dx = -2x-4 + c = -2/x⁴ + c

B. Square Roots

√x = x1/2 | ∛x = x1/3
Ex 1: ∫ 6√x dx = ∫ 6x1/2 dx = 4x3/2 + c = 4x√x + c
Ex 2: ∫ 10∛x dx = ∫ 10x1/3 dx = (15/2)x4/3 + c

C. Complex Fractions

Ex: ∫ (x² - 2x)/√x dx = ∫ (x3/2 - 2x1/2) dx = (2/5)x5/2 - (4/3)x3/2 + c

D. Expanding Brackets

Ex: ∫ x²(2x + 3) dx = ∫ (2x³ + 3x²) dx = x⁴/2 + x³ + c
Special: ∫ 1/x dx = ln|x| + c

5. Differential Equations

If dy/dx = f(x), then y = ∫ f(x) dx
Ex 1: dy/dx = 6√x → y = 4x3/2 + c
Ex 2: dy/dx = 2x - 3 → y = x² - 3x + c

6. Finding Curve Equations

Method:
  1. Integrate to get y = F(x) + c
  2. Substitute given point to find c
  3. Write final equation
Ex: dy/dx = 2x - 3, passes through (2, 3)
Step 1: y = x² - 3x + c
Step 2: 3 = 4 - 6 + c → c = 5
Answer: y = x² - 3x + 5

Second Derivatives

Integrate TWICE! Use c₁ for first, c₂ for second integration.
Ex: d²y/dx² = -12x + 4, at (3,34) gradient = -12
1st: dy/dx = -6x² + 4x + c₁ → c₁ = 30
2nd: y = -2x³ + 2x² + 30x + c₂ → c₂ = -20
Answer: y = -2x³ + 2x² + 30x - 20

7. Definite Integration

∫โ‚แต‡ f(x) dx = [F(x)]โ‚แต‡ = F(b) - F(a)
Steps:
  1. Integrate (no +c needed)
  2. Substitute upper limit b
  3. Substitute lower limit a
  4. Calculate: F(b) - F(a)
Ex 1: ∫₁³ x² dx = [x³/3]₁³ = 27/3 - 1/3 = 26/3
Ex 2: ∫₁² (x³ - x + 3) dx = [x⁴/4 - x²/2 + 3x]₁² = 8 - 11/4 = 21/4
Ex 3: ∫₀¹ x²(2x + 3) dx = [x⁴/2 + x³]₀¹ = 3/2

8. Areas Under Curves

Area = ∫โ‚แต‡ f(x) dx
⚠️ Important: Above x-axis = positive | Below x-axis = negative (use |result| for actual area)
Ex: Area bounded by y = 6x² - 4x + 2, x = -1, x = 1
Area = ∫₋₁¹ (6x² - 4x + 2) dx = [2x³ - 2x² + 2x]₋₁¹ = 2 - (-6) = 8 units²

9. Areas Between Two Curves

Area = ∫โ‚แต‡ [upper - lower] dx
Steps:
  1. Find intersections (set equal) → limits a, b
  2. Determine which is upper
  3. Calculate ∫ [upper - lower] dx
Ex: Between y = x + 7 and y = x² - 4x + 1
Intersections: x + 7 = x² - 4x + 1 → x = -1, 6
Area: ∫₋₁⁶ [(x + 7) - (x² - 4x + 1)] dx = ∫₋₁⁶ (-x² + 5x + 6) dx
= [-x³/3 + 5x²/2 + 6x]₋₁⁶ = 54 - (-19/6) = 343/6 units²

10. Composite Functions

∫ (ax + b)n dx = (1/a) × [(ax + b)n+1/(n+1)] + c
Quick Method: Power rule + divide by derivative of inside
Ex 1: ∫ √(3x + 1) dx = (2/9)(3x + 1)3/2 + c
Ex 2: ∫ 12/(3x - 4)² dx = -4/(3x - 4) + c

11. Quick Formula Reference

Function Integral
xnxn+1/(n+1) + c
k (constant)kx + c
1/xln|x| + c
√x(2/3)x3/2 + c
1/x²-1/x + c
1/√x2√x + c
(ax + b)n(1/a) × (ax + b)n+1/(n+1) + c

12. Exam Checklist

  • Power rule: add 1 to power, divide by new power, add c
  • ALWAYS add +c for indefinite integrals
  • Convert to powers: 1/xn = x-n, √x = x1/2
  • Definite integrals: NO +c, use limits [F(x)]โ‚แต‡ = F(b) - F(a)
  • Second derivatives: integrate twice, use c₁ and c₂
  • Areas: above x-axis = +, below = - (use absolute value)
  • Between curves: upper - lower, find intersections first
  • Composite: power rule ÷ derivative of inside
  • CHECK: Differentiate answer to verify!

13. Key Exam Examples

Q1: dy/dx = 6/(2x - 3)², passes (3, 5). Find x-intercept.
Integrate: y = -3/(2x - 3) + c
Find c: 5 = -3/3 + c → c = 6
Equation: y = -3/(2x - 3) + 6
x-intercept: 0 = -3/(2x - 3) + 6 → x = 7/4
Q2: ∫₀⁴ 3/√(2x + 1) dx
Integrate: 3√(2x + 1)
Limits: [3√(2x + 1)]₀⁴ = 9 - 3 = 6

๐Ÿ’ก Top 5 Tips

  1. Verify: Check by differentiating your answer
  2. Signs: Watch negative powers and coefficients
  3. Simplify first: Expand brackets, convert roots
  4. Draw: Sketch for area problems
  5. Show work: Get partial credit for method

๐ŸŽ“ Practice Makes Perfect!

Complete O Level Integration Guide

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