

Contents


Week 1: 
Indefinite Integrals: Primitive, basic properties of indefinite integrals. 
Week 2: 
Substitution method, integration by parts, 
Week 3: 
Integrals of rational functions. 
Week 4: 
Integrals of some irrational and transandental functions. 
Week 5: 
Definite Integrals: Riemann integral and its properties, some classes of integrable functions. 
Week 6: 
Mean Value Theorem of integral calculus, the fundamental therem of integral calculus. 
Week 7: 
Substitution in definite integral and integration by parts, functions of bounded variation. 
Week 8: 
Improper integrals, convergence of improper integrals, convergence tests. Midterm exam I 
Week 9: 
Applications of definite integrals: Area, arc length, volume, surface area. 
Week 10: 
Uniform convergence of sequences of functions, continuity of limit function. Uniform Cauchy sequence. Uniform convergence of series, termbyterm integration and differentiation. 
Week 11: 
Power series, radius of convergence, interval of convergence. Taylor series. 
Week 12: 
Weierstrass approximation theorem. Fourier series: Systems of orthogonal functions. 
Week 13: 
Pointwise Convergence of Fourier series, uniform convergence criteria, Bessel’s inequality, convergence in the mean, Parseval’s equality. 
Week 14: 
Fourier integrals and Fourier transformations. Midterm exam II 
Week 15*: 
 
Week 16*: 
Final exam 
Textbooks and materials: 

Recommended readings: 
G.M..FIKHTENGOL’TS “The fundamentals of Mathematical Analysis”, W.R.Parzgnski “Introduction to Mathematical Analysis”, Murray R.Spiegel “Advanced Calculus” 

* Between 15th and 16th weeks is there a free week for students to prepare for final exam.

