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# Cours scientifiques - MAA105 : Integral and Differential Calculus

## Descriptif

Integrals and differential calculus (MAA105) develop students's skills in crucial analytical tools, in particular integration theory. The approach to integration employed in this course is Riemann's integral, a foundational mathematical theory. This course also introduces students to two important and related topics covered in the Bachelor program: Taylor expansions (a tool for function approximation) and differential equations, which are required to understand basic physical problems (trajectories, populations, etc.)

The first part of this course focusses on the notion of Riemann integral. After introducing the notion of Riemann integrable function, we briefly discuss the basic properties of such functions. Next we present the classical methods for computing integrals (integration by parts, integration by substitution, integration of rational fractions, elementary abelian integrals…).
The second part is dedicated to the study of ordinary differential equations, mainly first order linear differential equations and linear systems of ODEs, with a special focus on linear differential equations with constant coefficients.

## Pour les étudiants du diplôme Bachelor of Science de l'Ecole polytechnique

Vous devez avoir validé l'équation suivante : UE MAA102

Numérique sur 20

## Pour les étudiants du diplôme Bachelor of Science de l'Ecole polytechnique

Le rattrapage est autorisé (Note de rattrapage conservée écrêtée à une note seuil de 11)
L'UE est acquise si Note finale >= 9
• Crédits ECTS acquis : 5 ECTS

La note obtenue rentre dans le calcul de votre GPA.

### Programme détaillé

• Integration of functions of one real variable, Riemann sums, fundamental theorem of analysis
• Integration by parts, changes of variables, calculus of primitive functions
• Decomposition of rational fractions
• Taylor formula with integral remainder; Taylor-Young formula, Landau notations o and O, computations and applications.
• ODEs.