Lec 15 | MIT 18.01 Single Variable Calculus, Fall 2007

The professor begins by transitioning from theoretical aspects of calculus, specifically the mean value theorem, to the practical application of integration. A new notation is introduced: differentials, denoted as 'dy' for a function 'y = f(x)', which is defined as 'f'(x) dx. This notation is attributed to Leibniz and is considered more effective than Newton's notation, with historical implications suggesting it contributed to the British lag in calculus development. The concept of infinitesimals is touched upon, and the differential dy is likened to the change in y, or delta y, with dx replacing delta x. The lecture aims to familiarize students with this notation, which will be prevalent throughout the course.

The professor illustrates the use of differentials with linear approximations, a concept previously covered using delta x and delta y. By introducing 'dx' as a small change in x and 'dy' as the corresponding change in y, the lecture simplifies the notation. An example calculation is performed to approximate 64.1^(1/3) using the function x^(1/3) and its differential, resulting in an approximate value of 4.002. The lecture emphasizes that while the notation has changed, the underlying mathematical principles remain consistent, highlighting the efficiency of Leibniz's notation in mathematical communication.

The lecture delves into the concept of antiderivatives, introducing the integral sign and the process of indefinite integration. The professor explains that the integral of a function g(x) with respect to dx results in an antiderivative, G(x), which is a function whose derivative is g(x). Several examples of integrals are provided, including the integral of sin x dx, which is -cos x + c, and the integral of x^a dx, which is (x^(a+1))/(a+1) + c, with a notable exception when a = -1. The lecture also addresses the ambiguity of antiderivatives due to the addition of an arbitrary constant 'c', emphasizing that this is the only source of ambiguity.

The professor continues to explore the concept of indefinite integrals, providing examples and highlighting special cases. The integral of dx/x is discussed, leading to the natural logarithm function, ln x, plus an arbitrary constant. The importance of considering both positive and negative values of x is noted, with the derivative of ln|x| for negative x being 1/x. Additional examples include the integral of sec^2 x dx, which is tan x + c, and the integral of √(1 - x^2) dx, which is sin^(-1) x + c. The lecture reinforces the idea that the antiderivative is unique only up to an arbitrary constant, a fundamental principle in calculus.

The professor emphasizes the importance of the fundamental theorem of calculus, which states that if the derivative of one function equals the derivative of another, then the two functions are identical up to an arbitrary constant. This theorem is crucial for the validity of calculus, as it ensures that the rate of change of a function determines the function itself, up to a constant. The lecture provides a proof for this theorem, highlighting its foundational role in calculus and its implications for the study of differential equations.

The professor introduces advanced techniques for integration, focusing on the method of substitution, which is particularly suited for complex integrals. An example is given where the integral of x^3 times (x^4 + 2)^5 dx is simplified by letting u = x^4 + 2, thus transforming the integral into a more manageable form. The lecture also introduces 'advanced guessing', a technique where one can predict the antiderivative based on experience and knowledge of differentiation formulas. Examples include integrals of e^(6x) dx and x e^(-x^2) dx, where the correct antiderivative is guessed and verified by differentiation.

The lecture discusses the uniqueness of antiderivatives and the use of trigonometric identities to verify different forms of the same integral. The professor presents the integral of sin x cos x dx and shows that it can be represented as either 1/2 sin^2 x + c or -1/2 cos^2 x + c, highlighting that the two forms are equivalent due to the Pythagorean identity sin^2 x + cos^2 x = 1. The importance of understanding that different-looking formulas can represent the same family of functions, differing only by a constant, is emphasized.

In the final part of the lecture, the professor tackles an integral involving a natural logarithm, ln x, by using the substitution method. The integral of 1/ln x dx is simplified by letting u = ln x, resulting in a straightforward integral of 1/u du, which integrates to ln u + c. The professor then converts back to the original variable, yielding ln(ln x) + c. The lecture concludes with a mention of upcoming topics, including differential equations and a review for an upcoming test, with a handout to be provided detailing the test contents.