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ANNOUNCER: The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. HERBERT GROSS: Hi, our topic today sounds a little bit naughty at first glance. It's called improper integrals. And rather than give a long speech about why they're called improper, let me try and start our lecture a little bit different from our usual approach, by giving you a little bit of a mathematical calculus riddle. Namely, let me just say this. We'll call our lecture today improper integrals, and we'll start off by saying find the flaw. I would like to integrate 'dx' over 'x squared' from minus 1 to 1. Using the first fundamental theorem of integral calculus, I say this is 'G of 1' minus 'G of minus 1' where 'G ...

ANNOUNCER: The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. HERBERT GROSS: Hi, today we start a new segment of our course. It could be entitled 'Techniques of Integration'. And a subtitle could be the difference between 'knowing what to do and knowing how to do it'. You see, essentially in this chapter, it shall be our aim to develop various recipes to evaluate the inverse derivative. So in fact, let's call it that. Some basic recipes. The idea is something like this. Let's start with an easier problem. Let's suppose we start with a differentiable function 'G'. Suppose, for example, that 'G' prime is equal to 'f'. Then in terms of the language of the indefinite integral, or the i...

ANNOUNCER: The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation, or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Hi. Our lesson today is called the 'Inverse Logarithm'. And what it will do, among other things, is give us a very nice chance to revisit inverse functions in general, only now applied specifically to the natural logarithm function that we talked about last time. So I call today's lesson, as I say, 'Inverse Logarithms'. And to see what's coming up here, simply recall that last time, we invented, so to speak, a new function called the 'natural log of x'. This had nothing to do with exponents. It was what? This was the function whose derivative with respect to 'x' was '1/x' and passed through the point (1 , 0). Tha...

ANNOUNCER: The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu. PROFESSOR: Hi. Our lecture for today I've entitled 'Logarithms Without Exponents'. And before I try to clarify that, let me make a couple of general remarks. First of all, we have now completed the rudiments, the basics so to speak, of both differential and integral calculus. Consequently, what our next ambition will be is to take these principles and apply them to special functions which are worthy of investigation. And you see, in connection with this, the first function that I've chosen to talk about is called the logarithm. Now the other interesting thing is that we are used to logarithms from high school where they were viewed as being a different kin...