Quick question about reverse chain rule

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etotheipi1

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I'm just starting to get to grips with integration and came across something pretty small but just wanted some clarification.

Sometimes when I am doing a simple integral such as the integral of (3x+5)2 dx, I can see that to apply the reverse chain rule you only need to remember to divide by the derivative of the inside function as well, namely 3. Thus you can write down 1/9 (3x+5)3.

However if the derivative of the inside function is not a constant, this method doesn't seem to work since you get a term with x in it multiplied by the original function which doesn't differentiate to the same thing. Would I be right in saying that you can only take the shortcut of dividing by the derivative if it is independent of x, and otherwise it is necessary to use u substitution?

Thank you and sorry if this sounds stupid!
 
"Reverse Chain Rule" is not a good expression or description. It's okay, I guess, but understanding what is happening is more important than the name we call it. You have observed, correctly, that this works only for constants. There must be other ways. Please show an example of one where you get something that is not a constant and let's see where the discussion goes.
 
What is the "reverse chain rule", as you have been taught it? I have seen the term used as a synonym for "u substitution" (which is itself a silly name, since the name of the variable you use is irrelevant!). You are taking it as something different.

I suspect that it is just a shortcut for substitution, which applies only when the "inner function" is linear (ax + b), as you suggest.

I want to see exactly what you were taught about it.
 
Thanks for your replies.

I have been taught that when you have a simple integral such as only having one inner function (e.g. (3x+5)2 or sin(2x) ), it is often best to think about what you need to differentiate to get this original function.

My teacher has been keen to make us better at integrating quickly by inspection without having to go through u substitution each time for easy integrals so the general strategy we have learned for these problems is to pretend you are applying the chain rule in reverse - that is integrate the outside function with respect to the inside function and then adjust for the chain rule by dividing by the derivative of the inner function. Taking sin(2x) as an example, I would say something along the lines of 'integrating the outside function of sin(2x) gives -cos(2x) but since this gives 2sin(2x) on differentiation the final answer needs to be -1/2 cos(2x)'.

This worked fine until I started wondering about things like (1+x2)8 in which case applying this reverse chain "rule" and dividing by 2x would yield a completely incorrect result.

I wanted to confirm that it only applies in cases where like you said the inner function takes the form (ax+b) meaning that when differentiated and rearranged for dx in u substitution the 1/a term can be taken out of the integral since it is a constant.

Thanks again for your help, I'm just starting to grapple with this sort of integration and want to make sure that the methods I am using are rigorous!
 
Okay, I see what you meant now. Integrating by inspection (when possible) is in fact what I do, and recommend to students; I just don't think of it as "reverse chain rule".

In fact, I do a little more than the chain rule in my thinking. I would look at (3x+5)2 and think, this is something squared, so the antiderivative ought to be something cubed. If I differentiate (3x+5)3, the chain rule gives me 3(3x+5)2 times 3; that's 9 times what I want, so I can divide by 9 to get the correct antiderivative. This saves me not only from having to formally do a substitution, but even having to memorize a formula for the antiderivative.

Of course, I keep in mind that if what I divided by was not a constant, then I couldn't do any of this, because the derivative of a quotient, in general, is not the quotient of the derivatives. It only works when the "something" (the u in a substitution) is a linear function. That's the part you were asking about, and you're right.
 
Dr.Peterson said:
Okay, I see what you meant now. Integrating by inspection (when possible) is in fact what I do, and recommend to students; I just don't think of it as "reverse chain rule".

In fact, I do a little more than the chain rule in my thinking. I would look at (3x+5)2 and think, this is something squared, so the antiderivative ought to be something cubed. If I differentiate (3x+5)3, the chain rule gives me 3(3x+5)2 times 3; that's 9 times what I want, so I can divide by 9 to get the correct antiderivative. This saves me not only from having to formally do a substitution, but even having to memorize a formula for the antiderivative.

Of course, I keep in mind that if what I divided by was not a constant, then I couldn't do any of this, because the derivative of a quotient, in general, is not the quotient of the derivatives. It only works when the "something" (the u in a substitution) is a linear function. That's the part you were asking about, and you're right.
Click to expand...

Thank you very much for your response, this clears things up very nicely!
 
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