Reverse chain rule integration

I was trying to integrate (2-1/x)^2

Why do you need to expand this before you integrate? When i first tried i got 1/3(2-1/x)^3*x^-2

This normally works. Why is this case an exception?

Reply 1

RDKGames

Study Forum Helper

Original post by stolenuniverse

I was trying to integrate (2-1/x)^2

Why do you need to expand this before you integrate? When i first tried i got 1/3(2-1/x)^3*x^-2

This normally works. Why is this case an exception?

The derivative of

2-\frac{1}{x}

is not a constant therefore the reverse chain rule does not work here - substitution is required instead, or expansion as you said.

Reply 2

Notnek

Original post by stolenuniverse

I was trying to integrate (2-1/x)^2

Why do you need to expand this before you integrate? When i first tried i got 1/3(2-1/x)^3*x^-2

This normally works. Why is this case an exception?

You can only use reverse chain rule when the stuff in the brackets is linear i.e. is of the form

ax+b

\displaystyle 2-\frac{1}{x}

is non-linear.

This is one of the worst understood topics in A Level so you're not alone :smile:

(edited 7 years ago)

Reply 3

DFranklin

Original post by notnek

You can only use reverse chain rule when the stuff in the brackets is linear i.e. is of the form

ax+b

\displaystyle 2-\frac{1}{x}

is non-linear.

This is one of the worst understand topics in A Level so you're not alone :smile:

Or to put it another way, it's just a straight out stupid thing to teach at A-level. TBH whenever a post contains the words "reverse chain rule" it feels it's almost the poster will be using it incorrectly.

Reply 4

Notnek

Original post by DFranklin

I think it's fine to show it after teaching substitution so students can see why reverse chain rule works (although the "topic" shouldn't even be given a name IMO). Integrals of the form

\int f(ax+b) \ dx

are very common in C4 exams so it's useful to be able to integrate them quickly without writing out a substitution.

The big problem is that reverse chain rule is right at the start of the integration chapter of the most popular textbook so most teachers teach it first. This always leads to problems when students try to use "reverse chain rule" for any integral.

(edited 7 years ago)

Reply 5

B_9710

You could only use reverse chain rule if you were integrating something like

\frac{1}{x^2}(1-1/x)^2

Reply 6

Notnek

Original post by B_9710

You could only use reverse chain rule if you were integrating something like

\frac{1}{x^2}(1-1/x)^2

In the Edexcel textbook, this is known as "integrating using standard patterns" :smile:

In the textbook "reverse chain rule" is only for integrating

f(ax+b)

.

I was never taught any of these names so it took a while for me to get used to them when I first started tutoring.

(edited 7 years ago)

Reply 7

B_9710

Original post by notnek

In the Edexcel textbook, this is known as "integrating using standard patterns" :smile:

Reverse chain rule is only for integrating

f(ax+b)

but this may differ depending on the textbook.

I was never taught any of these names so it took a while for me to get used to them when I first started tutoring.

Well yeah I say the 'reverse chain rule' just because it's familiar to the user, but I don't know what's actually 'standard' terminology.

Reply 8

tiny hobbit

Original post by DFranklin

TBH whenever a post contains the words "reverse chain rule" it feels it's almost certain the poster will be using it incorrectly.

My thoughts exactly. That's why I thought I'd read this thread.

Reply 9

DFranklin

Original post by notnek

I think it's fine to show it after teaching substitution so students can see why reverse chain rule works (although the "topic" shouldn't even be given a name IMO).Sure. But as you say, do it after you've done substitution and don't give it a name.

It's kind of funny how the people who can consistently use "the reverse chain rule" correctly never call it that, while nearly every time someone posts that "I used the reverse chain rule", the chances are they've done so incorrectly.

Of course, I'm probably falling into an "correlation does not imply causation" error myself here, but I really wish they wouldn't teach that particular piece of terminology...