INTEGRATION PM2
๐ INTEGRATION - Theory & Formulas
Cambridge AS & A Level Mathematics - Pure Mathematics 2
1. Introduction to Integration
Integration is the reverse process of differentiation.
Symbol: ∫ f(x) dx means "integrate f(x) with respect to x"
⚠️ Always add + c for indefinite integrals (constant of integration)
Example:
• Differentiate x² → get 2x
• Integrate 2x → get x² + c
• Differentiate x² → get 2x
• Integrate 2x → get x² + c
2. Integration of Exponential Functions
Basic Rules
∫ ex dx = ex + c
∫ e(ax+b) dx = (1/a) e(ax+b) + c
Example 1: ∫ e(3x) dx
a = 3, so answer = (1/3)e(3x) + c
Example 2: ∫ 6e(3x) dx
= 6 × (1/3)e(3x) + c = 2e(3x) + c
Example 3: Evaluate ∫₀² e(3x) dx
Step 1: Integrate: [(1/3)e(3x)]₀²
Step 2: Upper limit: (1/3)e⁶
Step 3: Lower limit: (1/3)e⁰ = 1/3
Answer: (1/3)(e⁶ - 1)
3. Integration of 1/(ax+b)
Basic Rules
∫ (1/x) dx = ln|x| + c
∫ 1/(ax+b) dx = (1/a) ln|ax+b| + c
⚠️ Always use |x| (absolute value) because ln is only defined for positive numbers
Example 1: ∫ (6/x) dx
Answer: 6ln|x| + c
Example 2: ∫ 4/(2x+1) dx
a = 2, so answer = 4 × (1/2)ln|2x+1| + c = 2ln|2x+1| + c
Example 3: Evaluate ∫₂³ 6/(2x-3) dx
= [3ln|2x-3|]₂³ = 3ln3 - 3ln1 = 3ln3
4. Integration of Trigonometric Functions
⚠️ These formulas ONLY work when x is in RADIANS!
Basic Rules
∫ cos x dx = sin x + c
∫ sin x dx = -cos x + c
∫ sec²x dx = tan x + c
General Rules
∫ cos(ax+b) dx = (1/a) sin(ax+b) + c
∫ sin(ax+b) dx = -(1/a) cos(ax+b) + c
∫ sec²(ax+b) dx = (1/a) tan(ax+b) + c
Example 1: ∫ sin(3x) dx
a = 3, so answer = -(1/3)cos(3x) + c
Example 2: ∫ 5cos(3x) dx
= 5 × (1/3)sin(3x) + c = (5/3)sin(3x) + c
Example 3: Evaluate ∫₀^(ฯ/4) (3 - 2sin2x) dx
= [3x + cos2x]₀^(ฯ/4) = (3ฯ/4 + 0) - (0 + 1) = 3ฯ/4 - 1
5. Advanced Trigonometric Integration
Use trigonometric identities to convert difficult functions into integrable forms
Key Identities
sin²x = (1 - cos2x)/2
cos²x = (1 + cos2x)/2
tan²x = sec²x - 1
⚠️ You CANNOT integrate sin²x or cos²x directly! Use identities first.
Example 1: ∫ sin²x dx
Step 1: Use sin²x = (1 - cos2x)/2
Step 2: = ∫ (1/2)(1 - cos2x) dx
Step 3: = (1/2)[x - (1/2)sin2x] + c
Answer: (x/2) - (sin2x)/4 + c
Example 2: ∫ cos²x dx
Use cos²x = (1 + cos2x)/2
Answer: (x/2) + (sin2x)/4 + c
Example 3: ∫ tan²x dx
Use tan²x = sec²x - 1
= ∫ (sec²x - 1) dx = tan x - x + c
Special Results
∫ sec x tan x dx = sec x + c
| Identity | Use for |
|---|---|
| sin²x = (1-cos2x)/2 | ∫ sin²x dx |
| cos²x = (1+cos2x)/2 | ∫ cos²x dx |
| tan²x = sec²x - 1 | ∫ tan²x dx |
6. The Trapezium Rule
Used to find approximate values when algebraic integration is not possible.
The Formula
∫โแต f(x) dx ≈ (h/2)[y₀ + yโ + 2(y₁ + y₂ + ... + yโ₋₁)]
where h = (b - a)/n
Easy way: Area ≈ (h/2) × [FIRST + LAST + 2(sum of all others)]
• n strips → n+1 ordinates (y-values)
• h = (b - a)/n
• Always make a table!
• h = (b - a)/n
• Always make a table!
Example: Use trapezium rule with 2 intervals to estimate ∫₁³ (x²+1) dx
Step 1: h = (3-1)/2 = 1
Step 2: x-values: 1, 2, 3
Step 3: y-values: 2, 5, 10
| x | 1 | 2 | 3 |
|---|---|---|---|
| y | 2 | 5 | 10 |
Step 4: Sum = 2 + 10 + 2(5) = 22
Step 5: Area ≈ (1/2) × 22 = 11
Over/Under-estimate
Concave curve (∩): Trapezium rule gives OVER-estimate
Convex curve (∪): Trapezium rule gives UNDER-estimate
Convex curve (∪): Trapezium rule gives UNDER-estimate
7. Summary of All Formulas
Exponential
∫ ex dx = ex + c
∫ e(ax+b) dx = (1/a)e(ax+b) + c
∫ e(ax+b) dx = (1/a)e(ax+b) + c
Logarithmic
∫ (1/x) dx = ln|x| + c
∫ 1/(ax+b) dx = (1/a)ln|ax+b| + c
∫ 1/(ax+b) dx = (1/a)ln|ax+b| + c
Trigonometric (RADIANS only)
∫ cos(ax+b) dx = (1/a)sin(ax+b) + c
∫ sin(ax+b) dx = -(1/a)cos(ax+b) + c
∫ sec²(ax+b) dx = (1/a)tan(ax+b) + c
∫ sin(ax+b) dx = -(1/a)cos(ax+b) + c
∫ sec²(ax+b) dx = (1/a)tan(ax+b) + c
Special Results
∫ sin²x dx = (x/2) - (sin2x)/4 + c
∫ cos²x dx = (x/2) + (sin2x)/4 + c
∫ tan²x dx = tan x - x + c
∫ sec x tan x dx = sec x + c
∫ cos²x dx = (x/2) + (sin2x)/4 + c
∫ tan²x dx = tan x - x + c
∫ sec x tan x dx = sec x + c
Quick Reference Table
| Function | Integral |
|---|---|
| ex | ex + c |
| e(3x) | (1/3)e(3x) + c |
| 1/x | ln|x| + c |
| 1/(2x+1) | (1/2)ln|2x+1| + c |
| cos x | sin x + c |
| sin x | -cos x + c |
| sec²x | tan x + c |
| sin²x | (x/2) - (sin2x)/4 + c |
| cos²x | (x/2) + (sin2x)/4 + c |
| tan²x | tan x - x + c |
๐ TIPS FOR SUCCESS
✓ Always add + c for indefinite integrals
✓ Check your work by differentiating
✓ Use RADIANS for trig functions
✓ Remember negative sign in ∫ sin x dx
✓ Use |x| when integrating 1/x
✓ Learn the identities
✓ Practice regularly!
⚠️ COMMON MISTAKES
❌ Forgetting + c
❌ Missing 1/a in formulas
❌ Forgetting negative in ∫ sin x
❌ Using degrees instead of radians
❌ Forgetting |x| in logarithms
❌ Integrating sin²x directly
❌ Wrong h formula in trapezium rule
❌ Not substituting both limits
