Calculus 1501A/B Lecture Notes - Lecture 3: Quadratic Function, Partial Fraction Decomposition, Abstract Algebra

• School: Western University
• Department: Calculus
• Course: Calculus 1501A/B
• Professor: Janusz Adamus

Document Summary

The rules of di erentiation give us an explicit algorithm for calculating derivatives of all ele- mentary functions, including trigonometric and exponential functions, as well as logarithms. By comparison, integration of elementary functions in general is a more di cult task. Z e x2 dx,z sin(x2) dx,z sin x x dx,z dx ln x recall that r dx cannot be expressed as elementary functions. To understand better this striking phenomenon, x = ln x, and we see that integration of a rational function leads to a transcendental function. In this lecture we discuss some more advanced techniques of integration. Let u = f (x) and v = g(x) be di erentiable functions with continuous derivatives. Then by the product rule: d(uv) = udv + vdu, or (2. 1) udv = d(uv) vdu. The antiderivative of the expression d (uv) is uv, and therefore, by integrating both sides of (2. 1) we obtain the formula of integration by parts: (2. 2)