# Integration by inspection

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Hi, I understand how to do integration by inspection but I just got confused by one example so I would appreciate it if somebody could help me.

I'll show what the solution says.

I just don't understand how

It feels like there is something extremely simple that I'm missing so any pointers on what happened during that step would be nice. Thanks in advance.

I'll show what the solution says.

I just don't understand how

It feels like there is something extremely simple that I'm missing so any pointers on what happened during that step would be nice. Thanks in advance.

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#2

(Original post by

Hi, I understand how to do integration by inspection but I just got confused by one example so I would appreciate it if somebody could help me.

I'll show what the solution says.

I just don't understand how

It feels like there is something extremely simple that I'm missing so any pointers on what happened during that step would be nice. Thanks in advance.

**Aiden223**)Hi, I understand how to do integration by inspection but I just got confused by one example so I would appreciate it if somebody could help me.

I'll show what the solution says.

I just don't understand how

It feels like there is something extremely simple that I'm missing so any pointers on what happened during that step would be nice. Thanks in advance.

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#3

**Aiden223**)

Hi, I understand how to do integration by inspection but I just got confused by one example so I would appreciate it if somebody could help me.

I'll show what the solution says.

I just don't understand how

It feels like there is something extremely simple that I'm missing so any pointers on what happened during that step would be nice. Thanks in advance.

The first integral gives sinx so it's just the second integral to worry about:

This is the one that can be done by inspection. Can you see how? Or is this the part where you're stuck?

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(Original post by

Do you understand what means? That reffers to the derivate of sinx. Maybe it will be clearer for you if it's kept as cosx:

The first integral gives sinx so it's just the second integral to worry about:

This is the one that can be done by inspection. Can you see how? Or is this the part where you're stuck?

**notnek**)Do you understand what means? That reffers to the derivate of sinx. Maybe it will be clearer for you if it's kept as cosx:

The first integral gives sinx so it's just the second integral to worry about:

This is the one that can be done by inspection. Can you see how? Or is this the part where you're stuck?

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(Original post by

Do you understand what means? That reffers to the derivative of sinx. Maybe it will be clearer for you if it's kept as cosx:

The first integral gives sinx so it's just the second integral to worry about:

This is the one that can be done by inspection. Can you see how? Or is this the part where you're stuck?

**notnek**)Do you understand what means? That reffers to the derivative of sinx. Maybe it will be clearer for you if it's kept as cosx:

The first integral gives sinx so it's just the second integral to worry about:

This is the one that can be done by inspection. Can you see how? Or is this the part where you're stuck?

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#6

(Original post by

Actually I think it is the bit that confuses me, I don't understand how to do that by inspection?

**Aiden223**)Actually I think it is the bit that confuses me, I don't understand how to do that by inspection?

Then the next question is: what differentiates to give . You should be able to get the general form and then adjust with a constant.

Over time you will recognise these forms and be able to see what they will integrate to.

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#7

**Aiden223**)

Actually I think it is the bit that confuses me, I don't understand how to do that by inspection?

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#8

(Original post by

Hi, I understand how to do integration by inspection but I just got confused by one example so I would appreciate it if somebody could help me.

I'll show what the solution says.

I just don't understand how

**Aiden223**)Hi, I understand how to do integration by inspection but I just got confused by one example so I would appreciate it if somebody could help me.

I'll show what the solution says.

I just don't understand how

As

the substitutes product in

the original integral

so the integral in correct form will be

that is you have to integrate wrt. sinx

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#9

**Aiden223**)

Hi, I understand how to do integration by inspection but I just got confused by one example so I would appreciate it if somebody could help me.

I'll show what the solution says.

I just don't understand how

Now, everywhere we have sin x, we want to treat it as a single object. To make that more obvious, let's replace it with U:

of course we can integrate this:

now, replace the U's with sin x's again: .

With practice, you can do the "treat sin x as a single object" thing without needing to replace it with a single letter, at which point you can go straight from to in one step.

(*) The d(sin x) notation is somewhat "dodgy", but if you are going to use it, it seems to me you must get rid of the dx (both because it "works" better but also because what you're "really" doing here is rewriting and the "dx" terms cancel (**)).

(**) Could the analysts stop screaming please - this is a methods question!

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#10

Note that in DFranklin's "replace sin(x) with U" step, he's *not* integrating by substitution, but simply formally replacing the term sin(x) with U.

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#11

(Original post by

Note that in DFranklin's "replace sin(x) with U" step, he's *not* integrating by substitution, but simply formally replacing the term sin(x) with U.

**Smaug123**)Note that in DFranklin's "replace sin(x) with U" step, he's *not* integrating by substitution, but simply formally replacing the term sin(x) with U.

replacing sinx with U and d(sinx) with dU.

With a substitution you have to change the variables/functions

with another variables/function and the dx differential with the proper differential

as integrating factor

If sinx=U then cosx dx =dU

or if there is no cosx as factor, then

x=arc sin U -> dx=1/\sqrt(1-U^2) dU

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#12

**Smaug123**)

Note that in DFranklin's "replace sin(x) with U" step, he's *not* integrating by substitution, but simply formally replacing the term sin(x) with U.

(Original post by

You can consider that it is a simple substitution , I think.

**ztibor**)You can consider that it is a simple substitution , I think.

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#13

For that matter, I've always disliked the phrase "integration by inspection", because one man's inspection might be another man's page of working, particularly if one person spends a lot of time dealing with certain integrals. Someone who often has to take moments of probability distributions might write:

"by inspection" but there's actually a *lot* of steps going on there (even if it's easy once you get the pattern).

For that matter, is solved instantly by the "substitution" , so if the integral is actually solvable, there's always *some* substitution that solves it "by inspection".

"by inspection" but there's actually a *lot* of steps going on there (even if it's easy once you get the pattern).

For that matter, is solved instantly by the "substitution" , so if the integral is actually solvable, there's always *some* substitution that solves it "by inspection".

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#14

(Original post by

For that matter, I've always disliked the phrase "integration by inspection", because one man's inspection might be another man's page of working, particularly if one person spends a lot of time dealing with certain integrals.

**DFranklin**)For that matter, I've always disliked the phrase "integration by inspection", because one man's inspection might be another man's page of working, particularly if one person spends a lot of time dealing with certain integrals.

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#15

(Original post by

For that matter, I've always disliked the phrase "integration by inspection", because one man's inspection might be another man's page of working,

**DFranklin**)For that matter, I've always disliked the phrase "integration by inspection", because one man's inspection might be another man's page of working,

(Original post by

What they don't teach at A-level, I believe, is the fact that "inspection" means "I spotted by magic that the answer is made substantially simpler by <this>, and I will now verify the magic step properly". As far as I remember, it's often simply taught as a synonym for "reverse chain rule".

**Smaug123**)What they don't teach at A-level, I believe, is the fact that "inspection" means "I spotted by magic that the answer is made substantially simpler by <this>, and I will now verify the magic step properly". As far as I remember, it's often simply taught as a synonym for "reverse chain rule".

I personally dislike both phrases "integration by inspection" and "reverse chain rule". And I'm pretty sure that neither existed in the (admittedly formal) textbooks I used to teach myself calculus many years ago.

"Integration by inspection" to me gives the false impression that there's some "magic box" operating, and that people with special skills of perception can "see" the answer, whereas weaker students cannot! I would always prefer (and always have done myself) to see students work through a substitution systematically (unless there's a very, very obvious differential hiding in there.), just to see that the student actually understands what's going on "under the hood". This is, perhaps, and old-fashioned view, and I note that the "young turks" on TSR seem to think that it's a weakness if students can't write down an integral "by inspection".

"Reverse chain rule" is even more of an anathema to me though. One of the weaknesses I see in modern students is an over-reliance on having some named rule for everything in existence, rather than trying to work things out from first principles, and one of the big weaknesses in the modern A level presentation of calculus is the failure to distinguish between differentiation, where a finite number of rules will always work (at least for any finite combination of functions encountered at A level), and integration, where it's easy to conjure up examples of simple composite functions that just can't be integrated! From what I've seen on TSR in recent years, use of the phrase "reverse chain rule" preys on

**both**weaknesses and leads many students to believe that with any sort of product or ratio or (especially) composition of functions they can just start integrating one part and dividing by another, Then they get confused when someone points out that integration doesn't work like this, because they want to fall back on the fact that something like (ax + b)^n seems to work when integrated using their technique!

It's very sad, I think, that students are no longer given an integral and expected to work out their own best approach to it using substitution and/or IBP as we used to have to do - every integration question now seems to be accompanied by the phrase "by using the substitution..." or "by integrating by parts..." to spell out how to do it!

That's my rant over for today

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#16

(Original post by

I'll throw in my twopence worth here - mainly because there hasn't been a more appropriate place to put this before, even though I've been thinking about this for a good few years!

**davros**)I'll throw in my twopence worth here - mainly because there hasn't been a more appropriate place to put this before, even though I've been thinking about this for a good few years!

I personally dislike both phrases "integration by inspection" and "reverse chain rule". And I'm pretty sure that neither existed in the (admittedly formal) textbooks I used to teach myself calculus many years ago.

"Integration by inspection" to me gives the false impression that there's some "magic box" operating, and that people with special skills of perception can "see" the answer, whereas weaker students cannot! I would always prefer (and always have done myself) to see students work through a substitution systematically (unless there's a very, very obvious differential hiding in there.), just to see that the student actually understands what's going on "under the hood". This is, perhaps, and old-fashioned view, and I note that the "young turks" on TSR seem to think that it's a weakness if students can't write down an integral "by inspection".

*was*probably a fair bit quicker at these kinds of integrals. But I'd still say that it's making mistakes that really costs me time with tough integrals, not being slow and methodical with the substitutions.

"Reverse chain rule" is even more of an anathema to me though.

One of the weaknesses I see in modern students is an over-reliance on having some named rule for everything in existence, rather than trying to work things out from first principles, and one of the big weaknesses in the modern A level presentation of calculus is the failure to distinguish between differentiation, where a finite number of rules will always work (at least for any finite combination of functions encountered at A level), and integration, where it's easy to conjure up examples of simple composite functions that just can't be integrated! From what I've seen on TSR in recent years, use of the phrase "reverse chain rule" preys on

**both**weaknesses and leads many students to believe that with any sort of product or ratio or (especially) composition of functions they can just start integrating one part and dividing by another,**wrong**.

(As you know), it's true that , and this seems to be the standard "correct" way of stating the inverse chain rule.

But somehow, most people seem to think you can divide both sides by g'(x) to get:

I'd love to know where this came from, because it's

**so**common I feel someone must be teaching it this way.

Incidentally, googling "reverse chain rule" doesn't leave me feeling that this is a particularly well founded technique. Looks like a bit of a "quirk" that has somehow become very popular with A-level / TSR students.

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