I have never taken away any points if a student forgot the +C on the exam, but…

[00:00:00]I have been wanting to make this video for a very long time because I want to say sometimes when we do an indefinite integral, if we forget the plus C at the very end, this right here is not that bad.

[00:00:13]Seriously.

[00:00:15]I've been a calculus teacher at a community college since 2013 and I can tell you I have never taken away points from a student's quiz or an exam that if they just forget the plus C at the very end.

[00:00:27]There are other things to worry about.

[00:00:30]For example, if you forget the differential here, this is a lot worse than not having a plus C at the very end.

[00:00:39]And you might be wondering what's wrong with the integral 1 over X being natural log absolute value of X plus C. Well, what if this right here we are in the U world?

[00:00:49]If this is a dU here, then this right here is horribly wrong.

[00:00:54]Yeah. And then in my opinion, the plus C really matters when we have an initial condition, especially when we wanted to solve a differential equation. So if you I really want the students to remember the plus C, I give them differential equations.

[00:01:10]Also, let's take a look at this one.

[00:01:14]What if we have the integral 1 over X dx and let's say the student doesn't know how to get the answer for that and they just maybe put on like a happy face and then they remember the plus C.

[00:01:28]Do you think that we should give them partial credit? It's of course not. I have never given any points if they just put on plus C on the test.

[00:01:36]So it's like, okay, I'm not going to take away points if you forget the plus C.

[00:01:42]In the meantime, I'm also not going to give you partial credit if you have the plus C as well.

[00:01:48]So yeah, I think that's how we can make this fair.

[00:01:52]Anyways, though, let me give you guys an example for this one right here real quick. Let's say the question is we want to integrate one over x times the square root of x minus one.

[00:02:04]First, we'll do u sub. Let u equal square root of x minus one.

[00:02:10]And we can isolate x first and then differentiate.

[00:02:13]So, square both sides and add one to both sides. x is equal to u squared plus one. And then differentiate this, we get 2 u du.

[00:02:24]Now, so far so good.

[00:02:27]Take this integral to the u world and we get the integral of one over x is right here.

[00:02:34]And then we have the u, right?

[00:02:37]And then dx is 2 u.

[00:02:41]And let's not forget the du.

[00:02:43]Okay, this is good because we see that the u and du cancel very nicely. So, we can put a two to the front and the integral one over x. And we see that's just two. And the integral one over x is ln absolute value of x.

[00:02:58]And yes, I remember the plus c. But how many points would you give me for this one right here?

[00:03:05]Not so many, right? Because from here to here is horribly wrong. Because we are in the u world. There's a du.

[00:03:14]There's a du.

[00:03:16]Right? There's a du right here.

[00:03:18]We cannot integrate x in the u world.

[00:03:22]So, what we have to do is change this, right? x is equal to u squared plus one.

[00:03:29]So, the answer is what? Inverse tangent of u from here.

[00:03:33]Right? So, two inverse tangent of u.

[00:03:37]And I know people like to say I don't put down the plus c right here. But I just like to put on the plus c at the very very end.

[00:03:46]I'm not completely done yet. This is two inverse tangent. u is square root of x minus one. I'll put that down right here.

[00:03:55]And then I'm all done. So I'll put a plus C right here.

[00:03:59]Yes.

[00:04:00]So that's it. And you might be wondering if the plus C doesn't really matter, then why am I wearing a plus C shirt? Well, of course, I also have a DX shirt right here.

[00:04:12]So yeah, don't worry about it.