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📐 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 ⭐ ∫ x n dx = x n+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 ...

Other note source 📐 UNDERSTANDING MEASUREMENT Cambridge IGCSE Mathematics - Grade 10 Complete Study Guide - Chapter 13 📏 13.1 UNDERSTANDING UNITS What Are Measurement Units? Measurement is how we describe the size, length, weight, or volume of objects. In everyday life, you measure things constantly - the distance to school, the amount of water in a bottle, or how heavy your backpack is. To communicate these measurements clearly, we use standardized units that everyone understands. The Metric System The metric system is the most widely used measurement system in the world. It's based on powers of 10, which makes conversions much easier than other systems. The metric system has three main categories: length , mass (weight) , and volume (capacity) . 🔑 Golden Rule of Conversion: To convert to a SMALLER unit → MULTIPLY To convert to a LARGER unit → DIVIDE 📏 LENGTH MEASUREMENTS Length measures how long or far something is. The basic unit i...

📚 Further Calculus - Complete Guide Pure Mathematics 3| Cambridge AS & A Level 8.1 Derivative of tan⁻¹(x) Understanding tan⁻¹(x): The inverse tangent function (arctan) reverses the tangent operation. If tan(θ) = x, then θ = tan⁻¹(x) Derivation Let y = tan⁻¹(x) Then: tan(y) = x Differentiate both sides: sec²(y) × dy/dx = 1 Therefore: dy/dx = 1/sec²(y) Using identity: sec²(y) = 1 + tan²(y) = 1 + x² Result: dy/dx = 1/(1 + x²) KEY FORMULA: d/dx[tan⁻¹(x)] = 1/(1 + x²) With chain rule: d/dx[tan⁻¹(f(x))] = f'(x)/(1 + [f(x)]²) Example 1: Differentiate tan⁻¹(3x) Solution: Let f(x) = 3x, so f'(x) = 3 d/dx[tan⁻¹(3x)] = 3/(1 + 9x²) Example 2: Differentiate tan⁻¹(√x) Solution: f(x) = x^(1/2), f'(x) = 1/(2√x) d/dx[tan⁻¹(√x)] = [1/(2√x)] / [1 + x] = 1/[2√x(1 + x)] 8.2 Integration of 1/(x² + a²) The Reverse Process: Since d/dx[tan⁻¹(x)] = 1/(1 + x²), we can integrate backwards! KEY FORMULAS: ∫ 1...

📐 VECTORS: Complete Study Guide Cambridge AS & A Level Mathematics - Pure Mathematics 3 1. INTRODUCTION TO VECTORS What is a Vector? A vector is a quantity that has two essential properties: 📏 Magnitude (Size/Length) 🧭 Direction Scalars vs Vectors Scalars (Number only) Vectors (Number + Direction) Temperature (25°C) Velocity (60 km/h North) Mass (5 kg) Force (10 N Downward) Distance (10m) Displacement (10m East) 2. DISPLACEMENT VECTORS (Section 9.1) 2.1 Writing Vectors Method 1: Column Vector ⎛ x ⎞ v = ⎜ y ⎟ ⎝ z ⎠ Where x is movement along the x-axis, y al...

Quiz:   Fullscreen Mode Other note source ⚛️ NUCLEAR MODEL OF THE ATOM 19.1 THE ATOM What is an Atom? ATOM = The smallest unit of a chemical element Structure of Atom ATOM = NUCLEUS + ELECTRONS Part Charge Location Mass Nucleus Positive (+) Center Almost all mass Electrons Negative (−) Orbit around nucleus Very small KEY FACT: Most of atom is EMPTY SPACE! Formation of IONS Neutral Atom: Number of Protons = Number of Electrons LOSE electrons → POSITIVE ION (+) GAIN electrons → NEGATIVE ION (−) Example: - Atom with 11 protons & 11 electrons = neutral - Loses 1 electron → 11 protons & 10 electrons = positive ion (+1) - Gains 1 electron → 11 protons & 12 electrons = negative ion (−1) Gold Foil Experiment (1911) Scientists: Geiger, Marsden, Rutherford Method: Shot alpha (α) particles at thin gold foil Results: Observation Conclusion Most α...