This course proposes like drank to make known to the student essential basic on the theory of integration of Riemann, as well as the techniques of the integral calculus. It also proposes to extend its knowledge to the generalized integrals and the convergence of these integrals.
Pedagogical Objectives: At the end of this course, the students must be able:
1. To know the direction and properties of the integral of Riemann of a numerical function on a limited closed interval of IR.
2. To know to bring back the calculation of such an integral to that of the primitives in the case of continuous functions.
3. To know to derive an integral function of its upper limit.
4. To know the formula of integration by parts as well as the formulas of change of variable and average and knowledge to apply these formulas.
6. To know the techniques of calculation of the primitives of the numerical functions.
7. To know methods of the approximate calculation of a definite integral (method of rectangles, of Simpson,….).
7. To recognize generalized integrals of first and second form and to know to calculate them when they converge.
8. To know general criteria of convergence of the generalized integrals and knowledge to apply.
9. To make the bond between convergence and absolute convergence of the generalized integrals as well as the concept of semi-convergence of these integrals and to know the rule of Abel.
Indefinite integrals, calculation of the primitives.
Definite integral: Riemann integral on a limited closed interval, Riemann-integrable functions, properties, theorem of the average, integral as function of its upper bound, fundamental theorem of the integral calculus, methods of integration.
Formula of Taylor with integral remainder.
Approximate calculation of a definite integral.
Generalized integrals: Criteria of convergence, absolute convergent, semi-convergent integrals, rule of Abel.
with an aim of better preparing the ground for the integral of Lebesgue later, one will be able to build the integral of Riemann starting from the functions in staircase and one will show that the integral thus to build is the limit of the sums of Riemann.
In the T.S. one will seek to especially cover all possible calculative techniques on the level of the primitives.