Lec 30 | MIT 18.01 Single Variable Calculus, Fall 2007

The paragraph introduces the topic of partial fractions, a method used to decompose rational functions into simpler fractions, which are easier to integrate. It emphasizes the reliability of this method for integration, despite its complexity. The professor also mentions the initial step of long division and the importance of the remainder's degree being less than the denominator's. The paragraph concludes with a reference to MIT OpenCourseWare, a free educational resource, and a call for donations to support its continuation.

This section delves deeper into the process of partial fractions, illustrating the method with an example of a polynomial divided by another polynomial. The professor explains the importance of factoring the denominator and the challenges associated with it, especially for higher degrees. The setup for partial fractions is described, including the pattern that emerges with linear factors and the corresponding terms needed for the numerator. The complexity of the process is highlighted, with the mention of 12 unknowns corresponding to the degrees of freedom in a polynomial of degree 11.

The professor acknowledges the complexity of the partial fractions method and its impracticality for hand calculations beyond simple cases. The paragraph discusses the process of separating the equation into 12 separate equations by taking coefficients of different powers of x. It also touches on the challenges of integration, even after simplifying the problem with the cover-up method, and the need for techniques like trigonometric substitution for certain integrals.

The script shifts focus to a new integration technique called 'integration by parts,' which is distinguished from partial fractions. The professor outlines the fundamental theorem of calculus and the product rule as the basis for this method. The integration by parts formula is introduced, and an example of integrating ln(x) is used to demonstrate its application, highlighting how it simplifies the process by making the function more manageable through differentiation.

The professor continues to demonstrate the power of integration by parts with the integral of (ln x)^2 dx. The process involves breaking down the integral into simpler components that can be more easily managed. The concept of a reduction formula is introduced, which allows for the expression of more complex integrals in terms of simpler ones, effectively reducing the problem step by step until it reaches a known integral.

This section explores the idea of reduction formulas in more depth, using the integral of ln(x^n) dx as an example. The professor shows how to apply integration by parts repeatedly to reduce the complexity of the integral. The process involves a series of steps that eventually lead to a simpler integral that can be solved directly, illustrating the method of induction and its power in solving integrals.

The professor provides additional examples of reduction formulas, including integrals involving exponential functions like e^x. The choice of u and v for the integration by parts process is discussed, emphasizing the importance of selecting functions that simplify upon differentiation. The section also mentions advanced guessing as an alternative method for tackling integrals and hints at the formulas that can be derived from integration by parts.

The script concludes with a practical application of integration by parts, calculating the volume of an 'exponential wine glass' formed by rotating the curve y = e^x around the y-axis. Two methods for setting up the integral are discussed: horizontal slices (method of disks) and vertical slices (method of shells). The professor demonstrates how the previously derived integral of (ln y)^2 can be used to find the volume using the method of disks, showcasing the utility of the techniques covered in the lecture.

In the final paragraph, the professor wraps up the discussion on the wine glass example by comparing the results obtained from the horizontal and vertical slicing methods. A correction is made to a previously missed minus sign in the calculation, emphasizing the importance of attention to detail in mathematical derivations. The paragraph reinforces the idea that the techniques taught can be applied to a wide range of integrals, providing a solid foundation for solving complex problems.