# GL(s,R)

## January 12, 2011

### A Calculus List

Filed under: High Effort/Low Payoff Ideas,Standard Based Grading — Adam Glesser @ 4:19 pm
Tags: ,

One of the advantages of my job is the incredible scheduling. We finished the fall semester the second week of December and my first class of the spring is next Tuesday! The downside is that this gives me way too much time to plot, scheme, doodle, dabble, think, rethink, and overthink. In the end, I usually settle on a plan that is far too ambitious, pedagogically impossible, philosophically suspect, and utterly indefensible. Thus, I bring you my plan for calculus this semester.

I read the wonderful article, Putting Differentials Back Into Calculus, which argues for using differentials in a way closer to their original creation than the way they are employed in modern textbooks.  As a huge fan of Thompson’s Calculus Made Easy, this suggestion didn’t seem half bad. Considering that only a fifth of my students are math majors and that, for the rest, using calculus outside of their physics class is unlikely, why not make things as easy as possible. I’m going to push to teach this same group next fall in calculus II, so the only teacher I can hurt is myself, right? But then I started thinking: the reason the differential approach will work so well with these students is that they will always be using differentiable functions. What reason is there to mention limits and continuity? These are technical issues that won’t help them at all in understanding calculus or how to apply it in their field of inquiry.

Oh dear, so here I am with essentially two months of material (this includes learning to differentiate any elementary function and using this to solve the standard problems). What will I do for the last month and half? I quickly remembered to add Taylor series because I love teaching that in calculus I. Then I added in the obligatory introduction to antiderivatives and integration. I even sprinkled in some partial differentiation at the end so that I could show the students the totally-awesome-implicit-differentiation-trick that would save them five minutes on the final exam. Grr…still two weeks left. These are precisely the two weeks that I usually spend on limits in the beginning. Now I remember why I always do this. It perfectly fills in the semester calendar. And then it hits me.

Any subject can be made repulsive by presenting it bristling with difficulties.
—Silvanus P. Thompson

Limits sure confuse the heck out of students. Why in the world are we leading calculus off with limits, especially to non-math majors? For the purpose of rigor?

You don’t forbid the use of a watch to every person who does not know how to make one? You don’t object to the musician playing on a violin that he has not himself constructed. You don’t teach the rules of syntax to children until they have already become fluent in the use of speech. It would be equally absurd to require rigid demonstrations to be expounded to beginners in the calculus.
—Silvanus P. Thompson

So I decide I just won’t do it.  No limits for us. We will just do some extra exploratory work. There is a great article on math in medicine we could read together. Okay, time for sleep.

But sleep does not come.

Toss.

Turn.

Toss.

Turn.

All right. All right. I’m up.

Why can’t the limits just die? Why do I feel the compulsion to put them back in? They’re like the tell-tale heart beating under my floor. What will become of my math majors if they don’t see limits? No, I can’t do that to them. What a cruel joke to play: send them to real analysis without having used limits. Back in they go. Of course, those biology majors are going to be completely turned off and once you lose them, they’re gone for good. Argh, out they go. On the other hand, if I get audited by the department, they are sure going to ask questions. With my review coming up, I can’t afford that kind of chatter. Put them back in…

…but later. Huh? Put them back in, but later. Yes, of course. Later. The course description says I need to cover limits, but it doesn’t say when! What if we introduced everything in a reasonable way and, only after the students know what is going on and why any of this is important, then showed them those funky limit do-hickies? Hmm. Interesting. And that is my explanation for the following calendar and skills list:

View this document on Scribd
View this document on Scribd

## January 7, 2011

### At long last…

Filed under: Music and Videos — Adam Glesser @ 5:00 pm

It only took a little over a year, but I finally got my collection of complex analysis lectures online. Sadly, two of the lectures mysteriously disappeared. In any case, as far as I know, this is the only (near) complete collection of videos on introductory complex analysis available online. Thanks to all the students who wrote me over the past year requesting I put these up. I wouldn’t have done it without your prompting.

For those who don’t know what complex analysis is, let me quickly summarize the idea as doing calculus but with functions of a complex variable. Cool things happen in this world that don’t normally happen. For instance, differentiable functions are automatically infinitely differentiable. Also, there is a certain theory called residue theory which enables one to use complex integration to solve real integration problems.

Enjoy!

## December 9, 2010

### All Hail Ailles

Filed under: High Effort/Low Payoff Ideas — Adam Glesser @ 5:28 pm
Tags: ,

This post is a sequel of sorts to my tricks of the trade post describing how to quickly compute the standard values of sine and cosine in the first quadrant. This is a very nice little exercise, known by quite a few high school teachers, but unknown to everyone I know personally. This lesson assumes that the student is aware of the basic facts from geometry about 30-60-90 and 45-45-90 right triangles as well as the Pythagorean theorem. If they know the definition of sine and cosine, you can go a little further at the end—this is my target audience. As a good preface to this, you might start with Kate Nowak’s post on these special right triangles. We start with the following rectangle:

Give this to the students (I do it in groups) and ask them find the length of each side and the measure of each angle. Simple arithmetic will take them here:

At this point, perhaps with some nudging, they apply the Pythagorean theorem to get at least two of the missing sides of the inner triangle.

The last side might be too intimidating for your students. If so, suggest they work on the angle problem. Those right triangles on the bottom should look mighty familiar…

My students got stuck here. However, I channeled my help less personality and let them struggle a bit. A few decided to work on the missing side again, but all eventually came around to noticing the angle on the bottom had to be 90 degrees—I must have said, “I don’t know. Why do you think it is a right angle?” half a dozen times that day.

It will not surprise you how few notice that the triangle is a 45-45-90 right triangle. It will surprise you how many don’t see that they can now use the Pythagorean theorem again to get the last side.

At this point, the students should have no problem sorting out the last two angles.

At the end, you might try asking them to compute the sine or cosine of $15^{\circ}$ and $75^{\circ}$. Normally, I teach these via the sum angle formulas, but this is a far more intuitive approach.

This rectangle was noticed by high school teacher Doug Ailles and is knows as the Ailles’ rectangle. Not too shabby.

## December 6, 2010

### Super-Sized, Circumcised, Circumscribed

Filed under: High Effort/Low Payoff Ideas — Adam Glesser @ 10:02 pm
Tags: ,

Today was supposed to be observation day. At least once per semester since I arrived at Suffolk, I’ve been observed in the classroom by one of the department’s senior faculty. Just so that you don’t get the wrong idea, this isn’t some kinky thing where I prance in front of the examiner in a Speedo. No, outside of Europe, nobody wants to see that kind of nonsense. The faculty actually want to watch how I teach a class.

While I always try to outwardly act as if this is no big deal, the thought of being observed gets me super excited. It gives me a really good excuse to spend that extra hour preparing a special lecture. With that in mind, this observation session came at a pretty awkward time: we just finished the course material and this is review week with the final coming next Tuesday. This is usually when we kick back, tighten up the few topics that are hurting everyone and watch Raiders of the Lost Ark. Instead, though, I decided to review old material by introducing something slightly new.

Enter the Extended Law of Sines

It always surprises me how few people have heard of the extended law of sines. In short, it says that the quantity $a/\sin(A)$ appearing in the law of sines is equal to the diameter of the circumcircle of the given triangle. If the radius of the circumcircle is $r$, then the theorem is usually written as:
$\dfrac{a}{\sin(A)} = \dfrac{b}{\sin(B)} = \dfrac{c}{\sin(C)} = 2r$
(where $a,b,c$ and $A,B,C$ represent the sides and angles of triangle, respectively, with $a$ opposite angle $A$, etc.). Aside from giving a geometric interpretation of the terms in the law of sines, the result is also quite useful (I’ll give a fairly involved exercise at the bottom where it is possible to use it).

I wanted the students to prove the result, but I really was not confident in their elementary geometry skills, so I set things up for them. I told them to start from the triangle and draw the circumcircle. Naturally, they didn’t know what I was talking about and wondered if I’d said what they thought I’d said. Planning for this, I started explaining out loud how you physically construct the circumcircle. Eyes glaze. Planning for this, I opened up handy-dandy mathopenref.com to show them what I can’t vividly describe verbally. Next came Geogebra where we found the circumcenter of the triangle and connected it to two of the vertices of the triangle. I now referenced the Central Angle Theorem which, again, nobody knew. Thankfully, Geogebra calculated the angles for us and since these students believe the convenient falsehood that one example proves that something always works, we were golden.

After connecting a few more dots, we were ready to prove the result. Here is the worksheet I gave them to work on in groups. It is nothing fantastic, but even some of the students whose attendance has been…spotty…worked out the answers. The beauty of the last question is that they need to shift gears and use the law of cosines before they can solve the problem.

Oh, and the professor who was coming to observe me? He didn’t show up. Go figure.

Exercise: With the notation as above, prove that the triangle is isosceles if and only if

$\dfrac{a+b+c}{2} = a\cos(B) + b\cos(C) + c\cos(A)$.

## December 4, 2010

### Evil SBG

Filed under: High Effort/Low Payoff Ideas,Standard Based Grading — Adam Glesser @ 5:40 pm

It’s been a while and I really ought to do some reflection on my first true semester of SBG. But, alas, the train ride is short and I wanted to throw something out there. I just finished proctoring the Putnam exam and had a wonderful idea for how to make SBG more evil.

You see, there are times when I really want to add weights to my topics. For instance, there is no way that finding the slope, y-intercept, etc. of a line should be worth as much in a precalculus class as graphing rational functions or using the law of cosines. It isn’t that I don’t think it is as important—it is probably more important in many cases—it is just that knowledge of lines (at that level) is considered remedial material, though for some it isn’t.

So how should I weight things? Should I do it by order of (my) perceived importance? How about by difficulty level—again from my point of view? I realized that while I may think I know the correct weightings, it is unlikely that I actually do. Consequently, I came up with this evil scheme. It isn’t really something serious, but I’m not completely joking either.

At the beginning of the semester, inform the students that each topic (or standard if you prefer) will be weighted by an amount inversely proportional—you’ll have to explain what that means—to the final average class score for that topic (or standard). The topic with the lowest class average score would then be weighted the highest. Those topics which everyone figures out would be weighted the least. This would give students (especially your top tier students) an incentive to attack those topics the whole class is failing. Potentially, students could try to game the system by learning the really hard stuff so well that it doesn’t count for so much. Hah! Now wouldn’t that be something?

## October 1, 2010

### Integration by Parts 3

Filed under: Tricks of the Trade — Adam Glesser @ 8:43 am
Tags: , ,

(with Professor Glesser)

In the first two installments of this series

we introduced integration by parts as a way to compute antiderivatives of a product of functions and we saw how certain integration by parts problems are handled more efficiently with the so-called tabular method (or, in Stand and Deliver, the “tic-tac-toe” method). In this post, we will consider the following question: As integration by parts requires the making of a choice—which is your u and which is your dv—how can we make this choice so that the resulting integral is easier to compute?

From the Mailbag

Über-reader CalcDave wrote in the comments to the last post in this series that,

I usually make a show of how sometimes the order does matter…That is, I’ll let $u = x^4$ and $dv = \sin(x)\ dx$ the first time and then go through it and say something like, “Well, that didn’t get us much of anywhere. What if we switch up our u and dv this time? Let’s let $u = \cos(x)$ and $dv = x^3$.” Then when you work it through, everything cancels out and we’re back to the original problem.

Indeed, Dave. Let’s take a look at what happens if we switch it up.

Egad, Dave is right. Since the product of the terms in the last line of the table is what we will need to integrate, doing it this way just makes things worse.  Ah, but what if we start with the cosine on the left and then switch it up? Oh, yeah, we’ll just get back what we started with. This suggests that we should always put a polynomials on the left so that it doesn’t go up in degree. It turns out that there are several examples where this is precisely the wrong thing to do. We implicitly saw this in the first post, but let me give you a couple of more explicit examples.

$\int x\sin^{-1}(x)\ dx$

If we split this up using our ‘rule’ to always put the polynomial on the left, then we are forced to integrate $\sin^{-1}(x)$. Let’s say you just happen to know the antiderivative of $\sin^{-1}(x)$ is $x\sin^{-1}(x) + \sqrt{1 - x^2} + C$ (I didn’t, although I can use integration by parts to figure it out!). You would now get:
and be forced to integrate the monstrosity on the right. Not for me thank you. However, if you put the $\sin^{-1}(x)$ on the left, we get:
and at the very least we have gotten rid of the $\sin^{-1}(x)$. In fact we have done more, but we’ll have to wait until the next post to resolve this.

Another example is $\int x\ln(x)\ dx$. Although we did integrate $\ln(x)$ in our first post, it gave an answer of $x\ln(x) - x + C$ and we don’t want to integrate that since it we don’t know how to integrate $x\ln(x)$ (in a future post, we will resolve this last problem directly). On the other hand, if we put the $\ln(x)$ on the left, the derivative will return $\frac{1}{x}$ and the natural logarithm is gone. So when does it pay to put the polynomial on the right? Whenever the derivative of the other function changes it into an algebraic function, it will be right to integrate the polynomial. Otherwise, you should differentiate the polynomial. If we also include trigonometric functions and exponential functions, the rule of thumb is:
Logarithms Inverse-Trig Algebraic Trig Exponential

This list represents a good order in which to choose your u in the following sense: if you have two functions, whichever comes first in the above list should be your u. Some people enjoy a good mnemonic to memorize the order. I’ve heard the following:

LIATE rule (or alL I ATE rule)

Lions In Africa Tackle Elephants

Liberals In America Typify Elitists

Little Indians Are Tiny Engines

Lets Integrate All The Equations

This says, for example, that when confronted with $\int \sin(x) e^x\ dx$, differentiate the $\sin(x)$ and integrate $e^x$.

Next Time

In our next segment, we will introduce the box method for handling several of the integrals left unsolved in this post.

## September 30, 2010

### You don’t know it until you know it

Filed under: Classroom Management,Standard Based Grading — Adam Glesser @ 8:36 pm

During my finite mathematics exam today, I had a little bit of time to ponder and I came up with two ideas worth exploring. Most likely, I shouldn’t implement them this semester, but I haven’t ruled it out. For posterity and potential discussion, I offer the following.

Filling the Empty Bucket

As a first approximation to the process of learning course content, imagine an empty box. This does not represent the knowledge of most of our students. Everyone has a few items in their box: some old postcards, two dead AA batteries, a dirty single sock, a couple of credit card applications, etc. The goal throughout the course is to fill that box. On occasion, a student might lose some things in their box after partying with Jack from next door and his cousin Daniels. However, with any reasonable effort, a student can keep the flow going into the box rather than going out. At the end of the course, their grade might simply be the percentage of the box that gets filled.

From a grading point of view, this suggests that my usual way of posting grades has a significant flaw. At any time, a student can see what their grade is, computed as a percentage of the possible points so far. Ah, there is the rub. A student might do well on the first quiz, regress a bit on the second, look at their grade and see 78%. Not great, but not too shabby considering they only need a C to graduate and they didn’t expect to be so close to a B-. Somehow, I’ve already lost. What if, instead of telling them how they’re doing relative to the available points, I gave them their grade in terms of the whole box? If my philosophical position is that their knowledge at the end of the course rather than their intermediate knowledge should determine their grade, then I should be telling them things like, “You have now mastered 21% of the material.”

There is an obvious problem with the original box metaphor: boxes are generally of constant size. It can’t all be about knowledge, though. I want their box to grow. This is not about content standards, but about them learning to think: the application of knowledge in logical and creative ways. How should I reflect that in the gradebook? Or is a one semester course insufficient to judge such growth? These aren’t new questions. David Cox explains things quite well here. I don’t have any new answer yet…but I’m still thinking.

The Exam Enigma

I mentioned in my last post how frustrated I am by the students adherence to the typical college way of studying. I had considered changing the names of exams to something less obvious; perhaps it might fool them. But who I am kidding. If I put it on the schedule, the students will figure it out. Maybe the whole problem is that I separate things in the first place. What if instead of treating these three exams as tests of retention, I simply clumped them in with the assessable standards portion of the grade? Right now, if a student does poorly on the standards, but studies hard and pulls off a good score on the exam, they get two grades: one says they don’t know what they’re doing, the other says they do. That makes no sense to me. I suggest (to myself) that I don’t differentiate quizzes and exams in the gradebook. The 75 minute exam should count no more than a daily quiz, except that the student will see more standards than usual on the test. This remands retention to the final exam.

Does it give the student an incentive to study throughout the term? Maybe it has the opposite effect. Maybe students will think nothing of the 75 minute exam and save up their study power for the final exam. It would be suicide, of course. They certainly won’t be able to cram the material in such a short time. Moreover, under this system, the final exam would be 20% and the standards + exams longer quizzes testing standards would be 70%. Some would still put all of their eggs in the final exam basket (another basket metaphor?), but that person won’t be helped by any system of assessment I try.

I was told today that one of my students was complaining about homework not being assessed. This student is having trouble because he or she isn’t doing any of the homework, nor asking about homework problems, nor writing me emails, nor attending office hours, etc. The student freely admits all of this, but, for not requiring homework, blames me. What is going on when people want to be treated like little children? Nothing is stopping the student from doing the homework problems but himself or herself. “But if I do the homework, how will I know if I got it right?” Aside from asking me, they could just look up the answers in the back of the book. I put up videos on youtube of me working the review problems. I put up links to videos of people explaining each standard. I beg for students to communicate with me outside of class. But it is my fault this student is struggling and because I don’t require him or her to do the homework? Well, sign up one more for the nanny state. Perhaps a class discussion is in order; let the kids doing the work hear from the kids not doing the work and see what they say to them.

## September 29, 2010

### An exam by any other name

Filed under: Uncategorized — Adam Glesser @ 7:10 pm

As I’ve mentioned in the past, I don’t have full control of my finite mathematics course. I asked for certain leeway in my grade breakdown, but had to settle on leaving the four class tests at 60% of the overall grade (in contrast to my precalculus where there are two tests making up 30% of the grade). Consequently, the students still treat my class as if nothing counts until exam week. Sure standards comprehension make up 30% of the overall grade, but they aren’t EXAMS!!!! How do I convince these kids that the typical college attitude will not cut it? They have an opportunity to spend four years living the life of the mind. Instead, they spend four years taking their mind off life. The night before a quiz, I stayed up late making screencasts of linear programming problems. One student told me how much he appreciated it, only he hadn’t had time to watch it because, “you know, the Yankees were on.” Guess how he did on the quiz?

Maybe I have to follow the rules and set my percentages like everyone else, but is there a chance that I could change the name (retention assesments?) and de-emphasize the importance of exams so that the day-to-day stuff gained relevance? Or are students socially constructed to work this way: slack, slack, slack, cram, rinse, repeat?

## September 24, 2010

### Integration by Parts 2

Filed under: Tricks of the Trade — Adam Glesser @ 8:28 am
Tags: , ,

Last time on

(with Professor Glesser)

we introduced integration by parts as an analogue to the product rule. We start this post with an example to show why the method can become tedious.

Consider

$\int x^4\sin(x)\ dx$

As there is a product of functions, this seems ideal for integration by parts. A question we will take up in our next post is which term we should look to differentiate (i.e., be our $u$) and which we should antidifferentiate (i.e., be our $dv$). For now, I will give you that a sound choice is

$u = x^4 \qquad dv = \sin(x)\ dx$

With this, we get

$du = 4x^3\ dx \qquad v = -\cos(x)$.

Using the integration by parts fomula:

$\int u\ dv = uv - \int v\ du$

we get

$\int x^4\sin(x)\ dx =-x^4\cos(x)-\int 4x^3(-\cos(x))\ dx$

Using linearity, we reduce the question to solving $\int x^3\cos(x)\ dx$.

Hold on, now. Is that really an improvement?

Yes, because the power of $x$ is smaller. But, I’ll grant you that life doesn’t seem much better. Essentially, we need to do integration by parts again. So, we rename things:

$u = x^3 \qquad dv = \cos(x)\ dx$
$du = 3x^2\ dx \qquad v = \sin(x)$

and we get

$\int x^3\cos(x)\ dx = x^3\sin(x) - \int 3x^2\sin(x)\ dx$

and after using linearity, we only need to compute $\int x^2\sin(x)$.

Before you get up and leave, notice that the power of $x$ is one less again.

Whoo-hoo. Yay, capitalism!

Seriously, each time we do this process, the exponent will decrease by one (since we are differentiating). So we “only” need to do it two more times.

You suck, Professor Glesser

Agreed. This is why it is nice to automate the process. I first learned this by watching Stand and Deliver over and over while in high school. I am not much of a fan of Battlestar Galactica (nerd cred…plummeting) and the few times I watched, I thought Edward James Olmos’ portrayal of William Adama was really flat; I thought Olmos was mailing in the performance. The most likely reason for my feelings? If you’ve never seen it, watch Stand and Deliver and Olmos’ portrayal of math teacher Jaime Escalante. Now that was a performance. Anyhow, here is the clip I watched incessantly.

I decided on a different notational scheme, but the method is the same. We make the following observation: when doing integration by parts repeatedly, the term that we differentiate will usually be differentiated again. That is, (abusing notation) the $du$ becomes our new $u$. If you like, the formula for integration by parts has us multiply diagonally left to right ($uv$) and then subtract the integral of the product left to right along the bottom ($-\int v\ du$):

The next iteration of integration by parts gives:

Essentially, this creates an alternating sum. In practice, it means we can set up the following chart where, going down, we differentiate on the left until we get $0$ and antidifferentiate on the right as many times as we differentiated.

Notice here that we are condensing quite a bit of notation with this method since we are no longer using the u, v, du, and dv notation. But, we are getting out precisely the same information. We draw diagonal left-to-right arrows to indicate which terms multiply and we superscript the arrows with alternating pluses and minuses to give the appropriate sign.

We don’t need to draw a horizontal arrow on the bottom since that would simply give us the antiderivative of $0 \cdot (-\cos(x)) = 0$. Following the arrows and taking account of signs, our antiderivative is

$-x^4\cos(x) + 4x^3\sin(x)+ 12x^2\cos(x)- 24x\sin(x)- 24\cos(x)+C$

Could you do that again?

Let’s try a different example, a little more complicated. Say we want to compute $\int (2x^2- 3x + 4)\cos(3x)\ dx$. We simply set up the chart where, going down, we differentiate on the left and antidifferentiate on the right:

and follows the arrows to get

$\frac{1}{3}(2x^2 - 3x+4)\sin(3x)+ \frac{1}{9}(4x-3)\cos(3x)- \frac{4}{27}\sin(3x)+C$
as the antiderivative for $\int (2x^2- 3x + 4)\cos(3x)\ dx$.

I think I need a break

Indeed. Next time we’ll take this a step further and show how to handle some situations where neither function is a polynomial. This will also bring up the question, again, about how to choose which function to differentiate and which to integrate.

## September 22, 2010

### Integration By Parts 1

Filed under: Tricks of the Trade — Adam Glesser @ 7:13 am
Tags: ,

This is the first in a series of posts on one of my favorite methods of antidifferentiation: integration by parts. I didn’t love it at first, but a little practice and a few tricks made me appreciate it. Teaching it, well, that is where the love affair begins.

(with Professor Glesser)

Let me assume that the reader is familiar with basic differentiation (including the product rule) and antidifferentiation of some basic elementary functions, i.e., the reader knows such facts as the power rule and how to antidifferentiate exponential functions as well as sine and cosine.

Integration by parts is an analogue to the product rule for derivatives (which tells you how to differentiate a product of functions). In the language of differentials, we have

$d(uv) = u\ dv + v\ du$

for functions $u$ and $v$ of some common variable, say $x$. Integrating both sides, we get

$uv = \int d(uv) = \int u\ dv + \int v\ du$.

The usual form of the integration by parts formula is now obtained by subtracting a term:

$\int u\ dv = uv - \int v\ du$.

Uh…What?

An example may be helpful. A canonical first example is $\int x\sin(x)\ dx$. The typical calculus student, fooled by the simplicity of the sum rule and not having the product rule in mind, will incorrectly assert $\int x \sin(x)\ dx = (\int x\ dx)(\int \sin(x)\ dx) = (\frac{1}{2}x^2)(-\cos(x)) = -\frac{1}{2}x^2\cos(x)$. Of course, differentiating shows that this answer is wrong. Why? Well, because antidifferentiation is additive but isn’t multiplicative.

So let’s try the integration by parts formula. We start by noting that $\sin(x)$ is the derivative of $-\cos(x)$, i.e., $d(-\cos(x)) = \sin(x)\ dx$. Consequently, we could write

$\int x \sin(x)\ dx = \int x\ d(-\cos(x))$.

We may now apply the integration by parts formula where $u = x$ and $v = -\cos(x)$. This gives

$\int x\ d(-\cos(x)) = -x\cos(x) - \int -\cos(x)\ dx = -x\cos(x) + \sin(x) + C$.

Could I see one more?

Sure, here is a less obvious example. Consider $\int \ln(x)\ dx$.

Wait, there is no product of functions.

There is a product; it is just a bit silly. You see $\ln(x) = \ln(x)\times 1$. Yes, it is one of those kinds of tricks. Now, I know that $1$ is the derivative of $x$ and so I can use the integration by parts formula with $u = \ln(x)$ and $v = x$. This gives:

$\int \ln(x)\ dx = x\ln(x) - \int x\ d(\ln(x)) = x \ln(x) - \int x \frac{1}{x}\ dx = x\ln(x) - \int dx = x\ln(x) - x + C$.

How do I keep everything straight?

A very common bookkeeping measure is to make a little table including $u, v, du$ and $dv$. For our first example, you would start with:

$u = x \qquad dv = \sin(x)\ dx$
$du = ? \qquad v = ?$

You then compute $du = 1\ dx = dx$ and $v = -\cos(x)$ to complete the table:

$u = x \qquad dv = \sin(x)\ dx$
$du = dx \qquad v = -\cos(x)$

You can then simply plug everything into the integration by parts formula.

This isn’t so bad. Why do you need multiple posts?

For those who don’t know the punchline, I won’t spoil it here. It suffices to say that there are some harder problems out there and there are some really efficient ways of handling these difficulties. Stay tuned!

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