Analyse I, partie 7: Intégration

An advanced mathematics course offered by EPFLx

Course Description

Welcome to "Analyse I, partie 7: Intégration," the final installment in the Analyse I series. This course focuses on the crucial concept of integration in calculus. Through comprehensive lectures and exercises, students will delve deep into the world of indefinite and definite integrals, exploring their properties, applications, and various integration techniques.

What Students Will Learn

  • Indefinite and definite integrals
  • Properties of definite integrals
  • The Mean Value Theorem for integrals
  • The Fundamental Theorem of Calculus
  • Various integration techniques
  • Integration of special functions
  • Generalized (improper) integrals
  • Integration of rational functions

Prerequisites

Students should have completed the previous parts of the Analyse I MOOC (parts 1-6). A strong foundation in calculus and mathematical reasoning is essential.

Course Coverage

  • Indefinite and definite integrals
  • Properties of definite integrals
  • Mean Value Theorem for integrals
  • Fundamental Theorem of Calculus
  • Integration techniques
  • Integration of trigonometric functions
  • Applications to numerical series
  • Integration of Taylor series
  • Integration of power series
  • Integration of piecewise continuous functions
  • Generalized (improper) integrals
  • Integration of rational functions

Who This Course Is For

  • Mathematics majors seeking to deepen their understanding of calculus
  • Science and engineering students who require advanced integration skills
  • Educators looking to enhance their knowledge of integration techniques
  • Self-learners passionate about advanced mathematics

Real-World Applications

  • Physics: Calculating work, energy, and fluid dynamics
  • Engineering: Analyzing structural loads and electrical circuits
  • Economics: Computing consumer and producer surplus
  • Statistics: Working with probability distributions
  • Computer Graphics: Rendering complex 3D shapes
  • Finance: Calculating present and future values of investments

Syllabus

Chapter 11: Indefinite and Definite Integrals

11.1 The indefinite integral
11.2 The definite integral
11.3 Properties of definite integrals
11.4 Mean Value Theorem
11.5 Fundamental Theorem of Calculus
11.6 First examples
11.7 Integration methods
11.7.5 Examples

Chapter 12: Integration (selected chapters)

12.1 Integration of Taylor series
12.2 Integration of power series
12.3 Integration of piecewise continuous functions
12.4 Generalized (improper) integrals
12.5 Examples of generalized integrals
12.6 Convergence of numerical series
12.7 Integration of rational functions
12.8 The general case

This course offers a comprehensive exploration of integration, equipping students with powerful mathematical tools applicable across various scientific and professional fields. Don't miss this opportunity to master one of the most fundamental concepts in calculus!