Is It Necessary to Substitute u in Integrals?

[1MileCrash]

I received no credit, resulting in an 84 for a few integral problems. I had correct final answers for everything.

When I confronted my professor about this, he said it was because I didn't actually put "u" and "du" into the integral. Is that really always necessary? Why actually put the u in if I'm just going to replace it back after, if I'm perfectly comfortable working with the actual statement (as most people should be!)? I do define u and du in order to "jot down" the function that I'm applying the theorem to.

From the test:

$\int\frac{3x}{\sqrt{x^2-15}}dx$

My work (first got rid of fraction)

$\int 3x(x^2-15)^\frac{-1}{2}dx$

let u = x^2 - 15; du = 2x dx


$\int \frac{3}{2}2x(x^2-15)^\frac{-1}{2}dx$

$\frac{3}{2} [\frac{(x^2 - 15)^\frac{1}{2}}{\frac{1}{2}}]$

$3 \sqrt{(x^2 - 15)} + C$



Now, I know that as with most "test complaint" threads there are going to be the guys that rush in here, gung ho, adamant that everything I did is completely wrong and that my professor is the most wonderful, compassionate, and blessed man to ever walk the Earth and that I should burn in the fiery pits of hell for questioning him or thinking that I actually know a little bit about math.

I work the problems better, and more efficiently this way. I would have to stop my "rhythm" to write something as useless to me as:

$\int u^\frac{-1}{2}\frac{3}{2}du$

$\frac{3}{2}\int u^\frac{-1}{2}du$

$\frac{3}{2} (2)u^\frac{1}{2}$

Just to put u back as x^2 - 15?

It just can't be that you guys feel the need to do that every time? Maybe for a long u that you just don't feel like writing out..

 

[pmsrw3]

People who do a lot of integrals often do the substitutions in their heads without writing them out explicitly, as you did. And I have no reason on Earth to believe that your prof is "the most wonderful, compassionate, and blessed man to ever walk the earth". I don't think you "should burn in the fiery pits of hell for questioning him". On the other hand, that doesn't actually seem to be the sanction on offer. Rather than burning in Hell, I gather that you lost 16 points out of 100 on an exam. Perhaps your opinion differs, but to me, losing 16 points seems a bit less severe than burning in Hell.

I think your prof has a right to define what counts as an acceptable answer on his exams. "Show your work" is a legitimate part of that. Especially in math, getting the right answer is not good enough. You have to say how you got there.

 

[1MileCrash]

I agree that showing the reasoning behind your answer is at least as important as the final result, actually more so in my opinion.

However, I don't see how my work leaves any gaps, and I did show all of my work. The "inverted chain rule (not sure what it's called!)" states that the integral of an inner function to an outer function multiplied by inner function derivative is the integral of the outer function to the inner function (I'm not sure how to type it out in english, but you know what I mean I'm sure) it doesn't require that I call the inner function some abstract variable u to work.

The only reasoning I see behind actually writing u in the place of the function itself is if the function is just annoying or long to write.. I didn't really do anything in my head.

 

[pmsrw3]

I think your prof has a right to define what counts as an acceptable answer on his exams.

 

[1MileCrash]

That has little to do with what's actually mathematically acceptable, which is what I'm much more concerned with.

 

[pmsrw3]

That has little to do with what's actually mathematically acceptable, which is what I'm much more concerned with.

"People who do a lot of integrals often do the substitutions in their heads without writing them out explicitly, as you did."

 

[thegreenlaser]

If the focus of the class (or at least that portion of it) was on doing u substitutions, then I'm not surprised you lost credit. I can't say I agree with you receiving no credit at all, but like pmsrw3 said, your prof gets to decide what's acceptable. I would bet that if you took a higher level course with the same prof, one where it's assumed that you know how to do u substitutions, then he probably would think your work was more than good enough.

When it's the focus of the class, I think it's actually a good thing, in a way, that he's being that strict. There are some people (I'm going to assume you're one of them) who either pick up on a concept quickly enough, or do enough practice problems that they can start to do steps in their head, and make shortcuts in their work. The problem is, there's a lot more people who THINK that they know a concept well enough to start taking shortcuts when they aren't. They aren't meticulous, and as a result they tend to make a lot of careless mistakes and they tend to have a lot of trouble when they have to use the technique later.

High school algebra is a good example of this. Pretty much no one in college would actually solve an equation by writing out their solution like this:

$$3x+2=8$$
$$\Rightarrow \ 3x+2-2=8-2$$
$$\Rightarrow \ 3x=6$$
$$\Rightarrow \ \frac{3x}{3}=\frac{6}{3}$$
$$\Rightarrow \ x=2$$

At very least, you would probably combine a few of those steps. In high school algebra, I remember my friend being really frustrated with having to write out steps like that, so he would try to do it all in his head. He was decently successful at it, but when algebra started coming up later in math/physics classes, he really struggled at it. He hadn't properly learned the techniques, and he was really shaky on the concept of doing the same thing to both sides of the equation. He would frequently make mistakes like subtracting 5 from the left side but not the right side, or similar. My friend doesn't represent every single person who makes shortcuts, but he represents a lot of people who think they understand stuff well enough to do it in their heads, when they actually don't. It's pretty much impossible for a prof to differentiate between the two, so it's good that he's being careful.

That's not to say you should burn in hell, or that you don't deserve any marks. Just try and see it from his perspective. Besides, I think that being super, even overly meticulous when you first learn a subject helps you to make a lot fewer mistakes when you start to do it in your head.

 

[Delusional]

I don't really understand why teachers insist on using explicit substitutions for totally banal integrals. The integral \begin{aligned}\int\frac{f'(x)}{2\sqrt{f(x)}}\;{dx} = \sqrt{f(x)}+\mathcal{C}\end{aligned} is one of the most basic standard integrals. It's like integration by parts when people unnecessarily define new variables.

 

[gb7nash]

If the focus of the class (or at least that portion of it) was on doing u substitutions, then I'm not surprised you lost credit. I can't say I agree with you receiving no credit at all

This

If the focus of this part of your class was u-substitution, write down all your steps. For a more extreme example, I could write:

$$\int x \sqrt{x^2+1} dx = \frac{1}{3} (x^2+1)^{\frac{3}{2}} + C$$

I know the answer, but the method is more important. You wrote down u and du correctly, which is good. However, if you don't write down all your steps, your teacher has no idea if you really know the material or just got lucky.

No credit is a little harsh, I personally would give partial credit, but by no means is your prof. obligated to.

 

[1MileCrash]

Besides, I think that being super, even overly meticulous when you first learn a subject helps you to make a lot fewer mistakes when you start to do it in your head.

I still have no idea how not replacing a mundane function with a variable, just to replace it back again is doing anything in my head.

If anything, putting u in just makes me constantly question if I'm multiplying by the right constant in order to make the second term equal to the derivative of the inner function, since it's no longer right there in front of me and I have to backtrack to a point before u was subbed in.

For example,

$\frac{3}{2}\int u^{\frac{-1}{2}}du$

Means: nothing.

While:

$\frac{3}{2}\int (x^2-15)^{\frac{-1}{2}} 2x dx$

Means: "right, by multiplying by 3/2, I have the derivative of the inner function as a multiplier, and I am still equal to the original problem."

I like to check my work as I work. Putting u in just makes me feel like I'm running blind.

 

[ArcanaNoir]

Unfortunately, things like this can happen in college. If you don't think it would change the letter grade you will ultimately get in the class, let it go. An 84 is very unlikely to kill what could be an A for the semester. From here on out, show every last shred of work that you possibly can. You should always do this on every exam for the rest of your life unless you are certain less is required. If you feel very strongly and want to make a big deal out of it, there's always administration, but odds there favor the professor and the amount of points you lost doesn't seem to warrant it.

 

[pmsrw3]

When it's the focus of the class, I think it's actually a good thing, in a way, that he's being that strict. There are some people (I'm going to assume you're one of them) who either pick up on a concept quickly enough, or do enough practice problems that they can start to do steps in their head, and make shortcuts in their work. The problem is, there's a lot more people who THINK that they know a concept well enough to start taking shortcuts when they aren't. They aren't meticulous, and as a result they tend to make a lot of careless mistakes and they tend to have a lot of trouble when they have to use the technique later.

It's a good point, and you have to remember that, although in your case there might really be no point in forcing you to do the integral through an explicit substitution, there might be others in the class who would just guess at the right answer (and verify it by differentiation). If he wants to make sure that they can use the technique, he really needs them to show it. And if he's going to require that of other members of the class, he also has to require it of you, or else he's liable to complaints of unfairness. (I'm a prof myself, and I can tell you, it hurts me to penalize mere carelessness when I know the student is a smart guy/gal and really gets it, but the fairness issue is real.)

For example,

$\frac{3}{2}\int u^{\frac{-1}{2}}du$

Means: nothing.

While:

$\frac{3}{2}\int (x^2-15)^{\frac{-1}{2}} 2x dx$

Means: "right, by multiplying by 3/2, I have the derivative of the inner function as a multiplier, and I am still equal to the original problem."

I like to check my work as I work. Putting u in just makes me feel like I'm running blind.

This sounds to me like an admission that you're not all there on integral substitutions :->

 

[thegreenlaser]

I still have no idea how not replacing a mundane function with a variable, just to replace it back again is doing anything in my head.

If anything, putting u in just makes me constantly question if I'm multiplying by the right constant in order to make the second term equal to the derivative of the inner function, since it's no longer right there in front of me and I have to backtrack to a point before u was subbed in.

For example,

$\frac{3}{2}\int u^{\frac{-1}{2}}du$

Means: nothing.

While:

$\frac{3}{2}\int (x^2-15)^{\frac{-1}{2}} 2x dx$

Means: "right, by multiplying by 3/2, I have the derivative of the inner function as a multiplier, and I am still equal to the original problem."

I like to check my work as I work. Putting u in just makes me feel like I'm running blind.

If you're not comfortable with u substitutions, it's probably something you should practice. When it comes down to it, doing it in your head is really a sort of 'guess and check' method. It may (and probably will) work in the vast majority of cases, but you should at least be comfortable with a u substitution. Have you done inverse trig substitutions yet? I personally don't know of anyone who consistently does those in their head. If u substitutions make you feel like you're 'running blind', then those are probably going to cause you a lot of trouble.

Another thing is, even if you never use it, it's a common method, and there's a good chance it'll come up when an professor or author is explaining another concept. If you're not comfortable with actually substituting the variable in, you're likely to have issues following the prof/author.

Again, I'm not saying you need to do it all the time, but the fact that carrying out the u substitution is making you feel uncomfortable isn't a good thing IMO.

EDIT: @ Delusional: Just because it seems simple to you now, doesn't mean it's simple to everyone in a freshman calculus class. Like I said before, there are a lot of people who think they get it when they really don't, so letting people cut corners right from the get go isn't generally a good thing.

 

[1MileCrash]

When it comes down to it, doing it in your head is really a sort of 'guess and check' method. It may (and probably will) work in the vast majority of cases, but you should at least be comfortable with a u substitution.

Please, enlighten me. What am I doing "in my head?"

And, enlighten me how replacing a function with a variable u and then replacing the u with the function later in the problem increases the "number of cases" the method works for?

Another thing is, even if you never use it, it's a common method, and there's a good chance it'll come up when an professor or author is explaining another concept. If you're not comfortable with actually substituting the variable in, you're likely to have issues following the prof/author.

Oh, I'm "comfortable" with it in the same way I'd be comfortable with renaming my functions for other inane operations for silly reasons. I'll do it if asked (explicitly), but I see no purpose unless the function is just long enough to where it's annoying to write out.

I'd compare it to renaming my two functions, smiley face and frowny face, before using the multiplication rule for derivatives.

This sounds to me like an admission that you're not all there on integral substitutions :->

Why? Because I'm not afraid to work with actual functions?

So, either I am doing it "in my head" and should stop "cutting corners" when I say it's a waste of time, and when I say I prefer to use actual functions it's because I'm "not all there" on integral substitutions. Really guys?

Have you done inverse trig substitutions yet? I personally don't know of anyone who consistently does those in their head. If u substitutions make you feel like you're 'running blind', then those are probably going to cause you a lot of trouble.

I'd have no problem using the method when it's more useful than using an actual function. But, I still don't know what I'm doing in my head.

 

[pmsrw3]

Why? Because I'm not afraid to work with actual functions?

I will quote you: "Putting u in just makes me feel like I'm running blind."

If that is not an admission that you're not comfortable with the method, I don't know what would be.

And now, I've had enough. You're sounding more and more like a whiny loser, and I'm sounding more and more like a pompous ***.

Feel free to have the last word.

EDIT: Interesting. PhysicsForum auto-censored me. A synonym for donkey was replaced by "***".

 

[1MileCrash]

I will quote you: "Putting u in just makes me feel like I'm running blind."

If that is not an admission that you're not comfortable with the method, I don't know what would be.

No, it's an admission that defining a function for a generic letter variable and replacing all instances of the function with it prevents me from viewing the actual function in my work, hence, "running blind."

I thought I was clear.

 

[nickalh]

Although,
A. I probably would have given 1MileCrash most or all of the credit.
B. I sympathize with you over the missed points.
C. It depends on how clear the professor was he specifically expected to put the u in.

Background:
I have taught simpler math classes, and tutored many calculus students but don't yet have the master's degree to teach calculus at an accredited school.

To respond to delusional,
it's just too easy to miss a coefficient and discover it half a page later, after looking for half an hour.

 

[thegreenlaser]

No, it's an admission that defining a function for a generic letter variable and replacing all instances of the function with it prevents me from viewing the actual function in my work, hence, "running blind."

I thought I was clear.

To be honest, this still doesn't sound like you're comfortable with the method. Why do you need to see the function explicitly written out? To me, that's definitely a sign of discomfort with the method. You say you like to check your work as you go, but that's just as easy (if not easier) with the substitution method.

I'm curious. How would you do this one, without using a substitution?

$$\int x \sqrt{x-1} \ dx$$

There's a very easy u substitution:

Spoiler

u=x-1

Solving it that way took me about a minute. It took me about five minutes to find a way to do it and avoid using a substitution, and that was even after knowing what the answer would look like. Sure, the second method is equally valid, but it's definitely a less efficient method in that case.

Even if you do find the above easier to solve without using a substitution, the "number of cases" in which substitutions work better is going to hit you pretty hard when you do inverse trig substitutions in calc II. There, you're either going to spend a lot of painful hours trying to do questions the long way, or you're just going to have to admit that renaming functions for inane and silly reasons is actually, on the contrary, quite a useful trick.

Again, I'm not saying what you did is invalid. I'm just not understanding why you're so determined to decry the substitution method.

 

[romsofia]

It helps when you get to trig integrals to actually write down the u in the integral, so I would suggest practicing now.