Chapter 3 (Part 2) Integration

Introduction to Integration

Integration is the inverse process of differentiation. If differentiation measures the rate of change, integration measures the accumulation of quantities. This chapter covers the fundamental concepts, formulas, and techniques of integration and differential equations.

📚 Key Concept

Anti-derivative: A function F(x) is called an anti-derivative of f(x) if F'(x) = f(x). The process of finding anti-derivatives is called integration.

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Anti-derivation Basics

Learn the fundamental concept of anti-derivatives and the relationship between differentiation and integration.

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Standard Integrals

Memorize essential integration formulas for algebraic, trigonometric, exponential, and other functions.

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Integration Methods

Master techniques like substitution, integration by parts, and partial fractions for solving complex integrals.

Anti-derivation and Differentials

Definition

The inverse process of differentiation is called anti-derivative or integration. The process of finding a function when its differential coefficient is known is called integration.

If F'(x) = f(x), then ∫f(x)dx = F(x) + C, where C is the constant of integration.

Differentials

If y = f(x), then differential dy = f'(x)dx. The differential represents the change in the function value for a small change in x.

Example: If y = x³, then dy = 3x²dx

💡 Memorization Tip

Remember: Differentiation and integration are inverse operations. The integral “undoes” what the derivative does, just like multiplication and division are inverse operations.

Important Notes

∫f'(x)dx = f(x) + C
∫f(x)dx represents the indefinite integral of f(x)
d/dx and ∫ are inverse operators of each other
If a function f(x) is continuous on [a, b], then it has an anti-derivative

Standard Integrals

1. Algebraic Functions

∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)

∫1/x dx = ln|x| + C

2. Trigonometric Functions

∫sin x dx = -cos x + C

∫cos x dx = sin x + C

∫sec² x dx = tan x + C

∫cosec² x dx = -cot x + C

∫sec x tan x dx = sec x + C

∫cosec x cot x dx = -cosec x + C

🎯 Quick Memorization

Trigonometric Integrals Pattern: The integrals of sin, cos, sec², cosec², sec tan, and cosec cot follow a simple pattern where the integral is the “co-function” (except with signs adjusted). Remember “sin→-cos”, “cos→sin”, “sec²→tan”.

3. Exponential Functions

∫eˣ dx = eˣ + C

∫aˣ dx = aˣ/ln a + C (a > 0, a ≠ 1)

4. Inverse Trigonometric Functions

∫1/√(1-x²) dx = sin⁻¹ x + C

∫1/(1+x²) dx = tan⁻¹ x + C

5. Hyperbolic Functions

∫sinh x dx = cosh x + C

∫cosh x dx = sinh x + C

Methods of Integration

1. Integration by Substitution

Used when the integral can be transformed into a standard form by substituting a new variable.

🔧 Problem Solving Tricks

∫f(ax+b)dx → Substitute u = ax+b
∫f'(x)/f(x)dx → Substitute u = f(x), gives ln|f(x)| + C
For √(a²-x²), use x = a sinθ
For √(a²+x²), use x = a tanθ
For √(x²-a²), use x = a secθ

2. Integration by Parts

Used for integrating products of functions: ∫u dv = uv – ∫v du

∫u v dx = u ∫v dx – ∫[u’ ∫v dx] dx

🔢 ILATE Rule for Choosing u

When using integration by parts, choose u in this order:

Inverse trigonometric functions
Logarithmic functions
Algebraic functions
Trigonometric functions
Exponential functions

Note: Some use “LIATE” which is the reverse order. Both work!

3. Integration by Partial Fractions

Used for rational functions (fractions where numerator and denominator are polynomials). Break complex fractions into simpler ones that can be integrated individually.

Definite Integrals

Integration with limits is called definite integral: ∫_a^b f(x) dx = F(b) – F(a)

∫_a^b f(x) dx = F(b) – F(a) where F'(x) = f(x)

Properties of Definite Integrals

∫_a^b f(x) dx = -∫_b^a f(x) dx
∫_a^b f(x) dx = ∫_a^c f(x) dx + ∫_c^b f(x) dx
∫_a^b f(x) dx = ∫_a^b f(a+b-x) dx
∫₀^a f(x) dx = ∫₀^a f(a-x) dx
∫_-a^a f(x) dx = 2∫₀^a f(x) dx if f(x) is even
∫_-a^a f(x) dx = 0 if f(x) is odd

📐 Geometric Interpretation

The definite integral ∫_a^b f(x) dx represents the signed area between the curve y = f(x), the x-axis, and the vertical lines x = a and x = b. Areas above the x-axis are positive, areas below are negative.

Differential Equations

An equation containing at least one derivative of a dependent variable with respect to an independent variable is called a differential equation.

Order and Degree

Order: The highest derivative in the equation
Degree: The power of the highest order derivative after removing radicals and fractions

Variable Separable Form

First order differential equations of the form: dy/dx = f(x)g(y) can be solved by separating variables:

∫(1/g(y)) dy = ∫f(x) dx

General vs Particular Solutions

General Solution: Contains arbitrary constants (equal to the order of the equation)
Particular Solution: Obtained by substituting specific values for the constants

🎓 Exam Strategy

For differential equations problems:

Identify the order and degree first
Check if it’s separable (dy/dx = f(x)g(y))
If separable, separate variables and integrate both sides
Use initial conditions to find particular solutions
Verify your solution by differentiating

Practice Quiz: Integration & Differential Equations

Test your knowledge with these 50 multiple choice questions. Click on your answer, then check the solution.

Question 1 of 50

Study Guidelines for Success

How to Master Calculus Integration

Memorize the basic formulas: Start with standard integrals of algebraic, trigonometric, and exponential functions.
Practice daily: Solve at least 5-10 integration problems every day to build muscle memory.
Understand the concepts: Don’t just memorize steps; understand why each integration technique works.
Learn to recognize patterns: Many integrals fit common patterns that can be solved with specific substitutions.
Master integration by parts: This is one of the most important techniques for solving complex integrals.
Check your answers: Always differentiate your result to verify it matches the original integrand.
Study in chunks: Break down your study sessions into focused topics (e.g., one hour on trigonometric integrals).
Use visualization: For definite integrals, sketch the area being calculated to build intuition.
Solve past papers: Practice with exam-style questions to understand the format and difficulty level.
Review mistakes: Keep an error log and review it regularly to avoid repeating the same mistakes.

⏰ Time Management for Exams

During exams: 1) Scan all questions first, 2) Start with questions you know best, 3) Allocate time based on marks, 4) Show your work for partial credit, 5) Leave time to review answers. For integration problems, if stuck, try a different method (substitution, parts, etc.).