# 📚 Grade 12 Mathematics: Antiderivatives (Integration)

---

## 📖 Table of Contents

1. [Introduction to Antiderivatives](#introduction-to-antiderivatives)
2. [Fundamental Concepts](#fundamental-concepts)
3. [Basic Integration Rules](#basic-integration-rules)
4. [Integration Techniques](#integration-techniques)
5. [Integration by Parts](#integration-by-parts)
6. [Trigonometric Integration](#trigonometric-integration)
7. [Integration by Substitution](#integration-by-substitution)
8. [Partial Fractions](#partial-fractions)
9. [Definite Integrals](#definite-integrals)
10. [Applications of Integration](#applications-of-integration)
11. [Important Formulas](#important-formulas)
12. [Practice Problems](#practice-problems)

---

## 🎯 Introduction to Antiderivatives

**Antiderivative** (or **Integration**) is the reverse process of differentiation. If we know the derivative of a function, integration helps us find the original function.

> 💡 **Key Insight:** If F'(x) = f(x), then F(x) is an antiderivative of f(x)

### Relationship with Differentiation

| **Differentiation** | **Integration** |
|:---:|:---:|
| d/dx[x²] = 2x | ∫ 2x dx = x² + C |
| d/dx[sin x] = cos x | ∫ cos x dx = sin x + C |
| d/dx[eˣ] = eˣ | ∫ eˣ dx = eˣ + C |

### Learning Objectives
By the end of this chapter, you will be able to:
- Understand the concept of antiderivatives
- Apply basic integration rules
- Use various integration techniques
- Evaluate definite integrals
- Solve real-world applications using integration

---

## 🔧 Fundamental Concepts

### 1. Definition of Antiderivative

**Definition:** A function F(x) is called an **antiderivative** of f(x) if:
```
F'(x) = f(x)
```

**Notation:**
- **Indefinite integral:** ∫ f(x) dx = F(x) + C
- **Definite integral:** ∫ₐᵇ f(x) dx

### 2. The Constant of Integration (C)

Since the derivative of a constant is zero, we must add an arbitrary constant C to indefinite integrals.

**Example:**
```
∫ 2x dx = x² + C
```

**Verification:**
```
d/dx[x² + C] = 2x + 0 = 2x ✓
```

### 3. Properties of Indefinite Integrals

#### Linearity Property
```
∫ [af(x) + bg(x)] dx = a∫ f(x) dx + b∫ g(x) dx
```

#### Constant Multiple Rule
```
∫ k·f(x) dx = k∫ f(x) dx
```

#### Sum/Difference Rule
```
∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx
```

---

## ⚡ Basic Integration Rules

### 1. Power Rule

**For n ≠ -1:**
```
∫ xⁿ dx = (x^(n+1))/(n+1) + C
```

**Examples:**
- ∫ x³ dx = x⁴/4 + C
- ∫ x⁻² dx = ∫ x⁻² dx = -x⁻¹ + C = -1/x + C
- ∫ √x dx = ∫ x^(1/2) dx = (2/3)x^(3/2) + C

### 2. Special Case: n = -1

```
∫ x⁻¹ dx = ∫ (1/x) dx = ln|x| + C
```

### 3. Exponential Functions

```
∫ eˣ dx = eˣ + C
∫ aˣ dx = aˣ/ln(a) + C   (a > 0, a ≠ 1)
```

### 4. Trigonometric Functions

| **Function** | **Antiderivative** |
|:---:|:---:|
| ∫ sin x dx | -cos x + C |
| ∫ cos x dx | sin x + C |
| ∫ sec² x dx | tan x + C |
| ∫ csc² x dx | -cot x + C |
| ∫ sec x tan x dx | sec x + C |
| ∫ csc x cot x dx | -csc x + C |

### 5. Inverse Trigonometric Functions

| **Function** | **Antiderivative** |
|:---:|:---:|
| ∫ 1/√(1-x²) dx | arcsin x + C |
| ∫ 1/(1+x²) dx | arctan x + C |
| ∫ 1/(x√(x²-1)) dx | arcsec x + C |

---

## 🛠️ Integration Techniques

### 1. Direct Integration

**Method:** Apply basic integration rules directly.

**Example:**
```
∫ (3x² - 2x + 5) dx = 3∫ x² dx - 2∫ x dx + 5∫ dx
                    = 3(x³/3) - 2(x²/2) + 5x + C
                    = x³ - x² + 5x + C
```

### 2. Integration by Substitution (u-substitution)

**When to use:** When the integrand contains a function and its derivative.

**Method:**
1. Choose u = g(x)
2. Find du = g'(x) dx
3. Substitute to get ∫ f(u) du
4. Integrate with respect to u
5. Substitute back u = g(x)

**Example:**
```
∫ 2x(x² + 1)⁵ dx

Let u = x² + 1
Then du = 2x dx

∫ 2x(x² + 1)⁵ dx = ∫ u⁵ du = u⁶/6 + C = (x² + 1)⁶/6 + C
```

### 3. Integration by Parts

**Formula:** 
```
∫ u dv = uv - ∫ v du
```

**LIATE Rule for choosing u:**
- **L**ogarithmic functions
- **I**nverse trigonometric functions
- **A**lgebraic functions
- **T**rigonometric functions
- **E**xponential functions

**Example:**
```
∫ x ln x dx

Let u = ln x,  dv = x dx
Then du = (1/x) dx,  v = x²/2

∫ x ln x dx = (ln x)(x²/2) - ∫ (x²/2)(1/x) dx
            = (x² ln x)/2 - ∫ x/2 dx
            = (x² ln x)/2 - x²/4 + C
```

---

## 🔍 Integration by Parts

### When to Use
- Product of two different types of functions
- When one function becomes simpler when differentiated

### Strategy Table

| **Type** | **Choose as u** | **Choose as dv** |
|:---:|:---:|:---:|
| ∫ x eˣ dx | x | eˣ dx |
| ∫ x sin x dx | x | sin x dx |
| ∫ ln x dx | ln x | dx |
| ∫ x² eˣ dx | x² | eˣ dx |

### 📝 Worked Examples

**Example 1:** ∫ x cos x dx
```
Let u = x,  dv = cos x dx
Then du = dx,  v = sin x

∫ x cos x dx = x sin x - ∫ sin x dx
             = x sin x - (-cos x) + C
             = x sin x + cos x + C
```

**Example 2:** ∫ x² eˣ dx
```
Let u = x²,  dv = eˣ dx
Then du = 2x dx,  v = eˣ

∫ x² eˣ dx = x² eˣ - ∫ eˣ (2x) dx
           = x² eˣ - 2∫ x eˣ dx

Now integrate ∫ x eˣ dx by parts again:
Let u = x,  dv = eˣ dx
Then du = dx,  v = eˣ

∫ x eˣ dx = x eˣ - ∫ eˣ dx = x eˣ - eˣ + C₁

Therefore:
∫ x² eˣ dx = x² eˣ - 2(x eˣ - eˣ) + C
           = x² eˣ - 2x eˣ + 2eˣ + C
           = eˣ(x² - 2x + 2) + C
```

---

## 📐 Trigonometric Integration

### 1. Powers of Sine and Cosine

#### Case 1: ∫ sinⁿ x cosᵐ x dx (n or m odd)
**Method:** Use substitution with the odd power.

**Example:** ∫ sin³ x cos² x dx
```
∫ sin³ x cos² x dx = ∫ sin² x cos² x sin x dx
                   = ∫ (1 - cos² x) cos² x sin x dx

Let u = cos x, then du = -sin x dx

= -∫ (1 - u²) u² du = -∫ (u² - u⁴) du
= -(u³/3 - u⁵/5) + C
= -cos³ x/3 + cos⁵ x/5 + C
```

#### Case 2: ∫ sinⁿ x cosᵐ x dx (both n and m even)
**Method:** Use half-angle formulas.

**Half-angle formulas:**
- sin² x = (1 - cos 2x)/2
- cos² x = (1 + cos 2x)/2

### 2. Tangent and Secant Integrals

| **Integral** | **Method** | **Result** |
|:---:|:---:|:---:|
| ∫ tan x dx | ∫ (sin x)/(cos x) dx, u = cos x | -ln\|cos x\| + C |
| ∫ sec x dx | Multiply by (sec x + tan x)/(sec x + tan x) | ln\|sec x + tan x\| + C |
| ∫ tanⁿ x dx | Use tan² x = sec² x - 1 | Various forms |

---

## 🔄 Integration by Substitution

### Types of Substitution

#### 1. Direct Substitution
**When:** Integrand contains f(g(x))·g'(x)

**Example:** ∫ (2x + 3)⁵ dx
```
Let u = 2x + 3, then du = 2 dx, so dx = du/2

∫ (2x + 3)⁵ dx = ∫ u⁵ (du/2) = (1/2)∫ u⁵ du
                = (1/2)(u⁶/6) + C = u⁶/12 + C
                = (2x + 3)⁶/12 + C
```

#### 2. Trigonometric Substitution
**For expressions involving:**

| **Expression** | **Substitution** | **Identity** |
|:---:|:---:|:---:|
| √(a² - x²) | x = a sin θ | sin² θ + cos² θ = 1 |
| √(a² + x²) | x = a tan θ | 1 + tan² θ = sec² θ |
| √(x² - a²) | x = a sec θ | sec² θ - 1 = tan² θ |

**Example:** ∫ 1/√(4 - x²) dx
```
Let x = 2 sin θ, then dx = 2 cos θ dθ
√(4 - x²) = √(4 - 4sin² θ) = 2√(1 - sin² θ) = 2 cos θ

∫ 1/√(4 - x²) dx = ∫ 1/(2 cos θ) · 2 cos θ dθ
                  = ∫ dθ = θ + C
                  = arcsin(x/2) + C
```

---

## 🧩 Partial Fractions

### When to Use
For rational functions where degree of numerator < degree of denominator.

### Types of Partial Fractions

#### Case 1: Linear Factors
```
P(x)/[(x-a)(x-b)] = A/(x-a) + B/(x-b)
```

#### Case 2: Repeated Linear Factors
```
P(x)/(x-a)ⁿ = A₁/(x-a) + A₂/(x-a)² + ... + Aₙ/(x-a)ⁿ
```

#### Case 3: Quadratic Factors
```
P(x)/[(x-a)(x²+bx+c)] = A/(x-a) + (Bx+C)/(x²+bx+c)
```

### 📝 Worked Example

**Example:** ∫ (2x + 1)/[(x - 1)(x + 2)] dx

**Step 1:** Set up partial fractions
```
(2x + 1)/[(x - 1)(x + 2)] = A/(x - 1) + B/(x + 2)
```

**Step 2:** Clear denominators
```
2x + 1 = A(x + 2) + B(x - 1)
```

**Step 3:** Solve for A and B
```
Let x = 1: 2(1) + 1 = A(3) + B(0) → A = 1
Let x = -2: 2(-2) + 1 = A(0) + B(-3) → B = 1
```

**Step 4:** Integrate
```
∫ (2x + 1)/[(x - 1)(x + 2)] dx = ∫ [1/(x - 1) + 1/(x + 2)] dx
                                = ln|x - 1| + ln|x + 2| + C
                                = ln|(x - 1)(x + 2)| + C
```

---

## 📏 Definite Integrals

### Definition
```
∫ₐᵇ f(x) dx = F(b) - F(a)
```
where F(x) is any antiderivative of f(x).

### Fundamental Theorem of Calculus

**Part 1:** If f is continuous on [a,b], then:
```
d/dx[∫ₐˣ f(t) dt] = f(x)
```

**Part 2:** If f is continuous on [a,b] and F is an antiderivative of f, then:
```
∫ₐᵇ f(x) dx = F(b) - F(a)
```

### Properties of Definite Integrals

1. **∫ₐᵃ f(x) dx = 0**

2. **∫ₐᵇ f(x) dx = -∫ᵇₐ f(x) dx**

3. **∫ₐᵇ f(x) dx + ∫ᵇᶜ f(x) dx = ∫ₐᶜ f(x) dx**

4. **∫ₐᵇ [f(x) ± g(x)] dx = ∫ₐᵇ f(x) dx ± ∫ₐᵇ g(x) dx**

5. **∫ₐᵇ k·f(x) dx = k∫ₐᵇ f(x) dx**

### 📝 Worked Examples

**Example 1:** ∫₀² (x² + 1) dx
```
∫₀² (x² + 1) dx = [x³/3 + x]₀²
                = (8/3 + 2) - (0/3 + 0)
                = 8/3 + 2 = 14/3
```

**Example 2:** ∫₀^(π/2) sin x dx
```
∫₀^(π/2) sin x dx = [-cos x]₀^(π/2)
                   = -cos(π/2) - (-cos(0))
                   = -0 + 1 = 1
```

---

## 📊 Applications of Integration

### 1. Area Under a Curve

**Area between f(x) and x-axis from x = a to x = b:**
```
A = ∫ₐᵇ |f(x)| dx
```

**Area between two curves f(x) and g(x):**
```
A = ∫ₐᵇ |f(x) - g(x)| dx
```

### 2. Volume of Revolution

#### Disk Method (rotation about x-axis)
```
V = π∫ₐᵇ [f(x)]² dx
```

#### Washer Method (rotation about x-axis)
```
V = π∫ₐᵇ [R(x)]² - [r(x)]² dx
```

#### Shell Method (rotation about y-axis)
```
V = 2π∫ₐᵇ x·f(x) dx
```

### 3. Arc Length

**Arc length of y = f(x) from x = a to x = b:**
```
L = ∫ₐᵇ √(1 + [f'(x)]²) dx
```

### 4. Average Value

**Average value of f(x) over [a,b]:**
```
f̄ = (1/(b-a))∫ₐᵇ f(x) dx
```

### 📝 Application Example

**Example:** Find the area between y = x² and y = 2x from x = 0 to x = 2.

**Solution:**
```
First, find intersection points:
x² = 2x → x² - 2x = 0 → x(x - 2) = 0
So x = 0 and x = 2

For 0 ≤ x ≤ 2: 2x ≥ x² (since 2x - x² = x(2 - x) ≥ 0)

Area = ∫₀² (2x - x²) dx
     = [x² - x³/3]₀²
     = (4 - 8/3) - (0 - 0)
     = 12/3 - 8/3 = 4/3
```

---

## 📋 Important Formulas

### Basic Integration Rules
```
∫ xⁿ dx = x^(n+1)/(n+1) + C  (n ≠ -1)
∫ 1/x dx = ln|x| + C
∫ eˣ dx = eˣ + C
∫ aˣ dx = aˣ/ln(a) + C
```

### Trigonometric Integrals
```
∫ sin x dx = -cos x + C
∫ cos x dx = sin x + C
∫ tan x dx = -ln|cos x| + C
∫ sec x dx = ln|sec x + tan x| + C
∫ sec² x dx = tan x + C
∫ csc² x dx = -cot x + C
```

### Inverse Trigonometric Integrals
```
∫ 1/√(1-x²) dx = arcsin x + C
∫ 1/(1+x²) dx = arctan x + C
∫ 1/(x√(x²-1)) dx = arcsec x + C
```

### Integration by Parts
```
∫ u dv = uv - ∫ v du
```

### Fundamental Theorem of Calculus
```
∫ₐᵇ f(x) dx = F(b) - F(a)
```

---

## 🎓 Practice Problems

### Problem 1: Basic Integration
Evaluate the following integrals:
1. ∫ (3x² - 2x + 1) dx
2. ∫ (√x + 1/x²) dx
3. ∫ (e^(2x) + sin x) dx

### Problem 2: Substitution
Evaluate:
1. ∫ x(x² + 1)³ dx
2. ∫ (2x + 1)√(x² + x + 1) dx
3. ∫ sin³ x cos x dx

### Problem 3: Integration by Parts
Evaluate:
1. ∫ x eˣ dx
2. ∫ x ln x dx
3. ∫ x² sin x dx

### Problem 4: Trigonometric Integration
Evaluate:
1. ∫ sin² x dx
2. ∫ tan³ x dx
3. ∫ sin x cos³ x dx

### Problem 5: Partial Fractions
Evaluate:
1. ∫ (x + 1)/[(x - 1)(x + 2)] dx
2. ∫ (2x + 3)/(x² + 3x + 2) dx

### Problem 6: Definite Integrals
Evaluate:
1. ∫₀¹ (x² + 2x) dx
2. ∫₀^(π/2) sin x cos x dx
3. ∫₁² (1/x + eˣ) dx

### Problem 7: Applications
1. Find the area between y = x² and y = 4 from x = 0 to x = 2
2. Find the volume when y = √x is rotated about the x-axis from x = 0 to x = 4
3. Find the average value of f(x) = x² over [0, 3]

---

## 🧠 Problem-Solving Strategies

### General Approach

1. **🎯 Identify the Type of Integral**
   - Basic polynomial?
   - Trigonometric function?
   - Product of functions?
   - Rational function?

2. **📝 Choose the Appropriate Method**
   - Direct integration
   - Substitution
   - Integration by parts
   - Partial fractions
   - Trigonometric substitution

3. **🔍 Apply the Technique**
   - Follow the method step-by-step
   - Be careful with algebra
   - Don't forget the constant of integration

4. **✅ Verify Your Answer**
   - Differentiate your result
   - Check if you get the original integrand
   - Verify definite integral bounds

### Method Selection Guide

| **Integrand Type** | **Method** | **Example** |
|:---:|:---:|:---:|
| Polynomial | Direct | ∫ x³ dx |
| f(g(x))·g'(x) | Substitution | ∫ 2x(x²+1)³ dx |
| Product of different types | Integration by parts | ∫ x eˣ dx |
| Rational function | Partial fractions | ∫ 1/(x²-1) dx |
| √(a²-x²), etc. | Trigonometric substitution | ∫ 1/√(1-x²) dx |

### Common Mistakes to Avoid

❌ **Forgetting the constant of integration**
❌ **Incorrect application of integration by parts**
❌ **Wrong substitution choice**
❌ **Algebraic errors in partial fractions**
❌ **Incorrect evaluation of definite integrals**

---

## 🎯 Key Takeaways

### Remember These Concepts:
- **Integration** is the reverse of differentiation
- **Constant of integration** is essential for indefinite integrals
- **Substitution** works when you can identify u and du
- **Integration by parts** follows the LIATE rule
- **Partial fractions** decompose rational functions
- **Definite integrals** give numerical values

### Success Tips:
- 📖 **Master basic integration rules** first
- 🎯 **Practice pattern recognition** for choosing methods
- 🧮 **Check your work** by differentiating
- 📊 **Understand geometric interpretations**
- 🔍 **Work systematically** through complex problems

---

## 📚 Additional Resources

### Recommended Study Materials:
- Calculus textbooks with integration chapters
- Online integration calculators for verification
- Practice problem sets with step-by-step solutions
- Video tutorials on integration techniques

### Common Integration Tables:
- Standard integral formulas
- Trigonometric integral formulas
- Exponential and logarithmic integrals
- Inverse trigonometric integrals

---

*📝 These notes provide a comprehensive foundation for understanding antiderivatives and integration in Grade 12 Mathematics. Regular practice with various types of integrals will build proficiency in this essential calculus topic.*

---

**📞 Need Help?**
- Review the method selection guide
- Practice with simpler problems first
- Check your algebra carefully
- Use online tools for verification

**🌟 Good luck with your calculus studies!**