Edexcel A Level Maths Integration Explained: Complete Guide

Integration is the reverse of differentiation and a fundamental concept in Edexcel A Level Mathematics. This complete guide explains what integration is, how to integrate different functions, and applications including finding areas and volumes.

What is Integration?

Integration is the reverse process of differentiation. It allows us to:

• Find the area under a curve
• Find volumes of revolution
• Solve differential equations
• Reverse differentiation

Notation:

• ∫ f(x) dx (integral sign)
• ∫ means “integrate” or “find the integral of”
• dx means “with respect to x”
• • C (constant of integration)

Basic Integration Rules

Power Rule

• ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (where n ≠ -1)
• Add 1 to power, divide by new power
• Add constant of integration

Examples:

• ∫ x³ dx = x⁴/4 + C
• ∫ x⁵ dx = x⁶/6 + C
• ∫ x⁻² dx = x⁻¹/(-1) + C = -1/x + C

Special Case:

• ∫ 1/x dx = ln|x| + C (not power rule)

Constant Rule

• ∫ k dx = kx + C (where k is constant)
• Constants integrate to kx

Examples:

• ∫ 5 dx = 5x + C
• ∫ -3 dx = -3x + C

Sum/Difference Rule

• Integrate each term separately
• ∫ [f(x) ± g(x)] dx = ∫ f(x) dx ± ∫ g(x) dx

Example:

• ∫ (x³ + 2x² - 5x + 3) dx
• = x⁴/4 + 2x³/3 - 5x²/2 + 3x + C

Advanced Integration Techniques

Integration by Substitution

• Replace part of function with u
• Find du/dx
• Substitute and integrate
• Replace u back

Example:

• ∫ (3x + 2)⁵ dx
• Let u = 3x + 2, so du/dx = 3, dx = du/3
• ∫ u⁵ (du/3) = (1/3) ∫ u⁵ du = (1/3)(u⁶/6) + C
• = (3x + 2)⁶/18 + C

Integration by Parts

• Formula: ∫ u dv = uv - ∫ v du
• Choose u and dv carefully
• Differentiate u, integrate dv
• Apply formula

Example:

• ∫ x eˣ dx
• Let u = x, dv = eˣ dx
• Then du = dx, v = eˣ
• ∫ x eˣ dx = x eˣ - ∫ eˣ dx = x eˣ - eˣ + C = eˣ(x - 1) + C

Partial Fractions

• Split fraction into simpler parts
• Integrate each part separately
• Useful for rational functions

Definite Integration

What is Definite Integration?

• Integration with limits (upper and lower bounds)
• Gives numerical answer (not function)
• No constant of integration needed
• Represents area under curve

Notation:

• ∫[a to b] f(x) dx
• a = lower limit, b = upper limit
• Evaluate at b, subtract value at a

Example:

• ∫[0 to 2] x² dx
• = [x³/3][0 to 2]
• = (2³/3) - (0³/3)
• = 8/3 - 0 = 8/3

Fundamental Theorem:

• ∫[a to b] f(x) dx = F(b) - F(a)
• Where F(x) is antiderivative of f(x)

Applications of Integration

Area Under Curve

• ∫[a to b] f(x) dx gives area between curve and x-axis
• If curve below axis, area is negative
• Total area = sum of absolute values

Area Between Curves

• ∫[a to b] [f(x) - g(x)] dx
• Where f(x) is upper curve, g(x) is lower curve
• Find intersection points first

Volume of Revolution

• Rotate curve around x-axis
• V = π ∫[a to b] [f(x)]² dx
• Square the function, multiply by π

Average Value

• Average = (1/(b-a)) ∫[a to b] f(x) dx
• Mean value of function over interval

Common Integrals

Standard Integrals:

• ∫ eˣ dx = eˣ + C
• ∫ 1/x dx = ln|x| + C
• ∫ sin x dx = -cos x + C
• ∫ cos x dx = sin x + C
• ∫ sec² x dx = tan x + C

Learn These:

• Essential for A Level
• Appear frequently
• Save time in exams
• Check by differentiating back

Step-by-Step Problem Solving

Problem 1: Find ∫ (2x³ - 3x + 5) dx

1. Integrate each term: 2x⁴/4 - 3x²/2 + 5x
2. Simplify: x⁴/2 - 3x²/2 + 5x
3. Add constant: x⁴/2 - 3x²/2 + 5x + C

Problem 2: Find area under y = x² from x = 0 to x = 3

1. Set up integral: ∫[0 to 3] x² dx
2. Integrate: [x³/3][0 to 3]
3. Evaluate: (3³/3) - (0³/3) = 9 - 0 = 9
4. Area = 9 square units

Problem 3: Find volume when y = x² rotated around x-axis from x = 0 to x = 2

1. V = π ∫[0 to 2] (x²)² dx = π ∫[0 to 2] x⁴ dx
2. Integrate: π [x⁵/5][0 to 2]
3. Evaluate: π [(2⁵/5) - (0⁵/5)] = π(32/5) = 32π/5

Common Mistakes to Avoid

1. Forgetting Constant of Integration

• Always add + C for indefinite integrals
• Not needed for definite integrals
• Check your work

2. Not Adding 1 to Power

• Always add 1, then divide
• Check: differentiate back should give original

3. Definite Integral Errors

• Evaluate at upper limit, subtract lower limit
• Don’t forget to substitute limits
• Check arithmetic

4. Sign Errors

• Be careful with negative signs
• Check when integrating negative terms
• Verify your answer

Related Resources

• A Level Maths Differentiation Explained
• A Level Maths Past Papers
• A Level Maths Tutor

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Understanding integration is essential for A Level Maths. Practice regularly and seek help when needed to master this fundamental concept.