Integrate In Seconds: The Reverse Chain Rule
Is there a way to streamline the process of integration by substitution? In comes the Reverse Chain Rule to save our precious time.
But first, before we can reverse it, we should refresh our memory of how the chain rule works.
Chain Rule
The Chain Rule is an essential tool in differentiating otherwise hopelessly complicated functions. Like this one: f(x) =
If there existed a mathematical hell, one of the punishments would likely be having to expand the function into its glorious 46th degree polynomial form and using the power rule to differentiate each term.
Fortunately, the chain rule comes to our rescue.
Let’s imagine the chain rule as a machine that accepts the input function y = f(u), which is a composition of the function f and u.
y will be fed into an operator which differentiates it with respect to u. The inner function u is separated to be fed into a different operator, which differentiates it with respect to x.
The outputs of the two operators, dy/du and du/dx, are then multiplied to produce dy/dx.
Returning to our original function,
This is much more agreeable than the 45 term expression we would have gotten from differentiating the expanded polynomial.
What if you were asked to integrate?
At first, it seems like an extremely ugly integral. But from what we have just done with the chain rule, we know the integrand is the derivative of our first function. So we can just exploit the fact that integration is the reverse of differentiation and say the integral must be the original function plus some arbitrary constant.
We can see a pattern emerging that will help us integrate composite functions just as the chain rule allows us to differentiate.
Reverse Chain Rule
If the integrand is a product of a composite function f(u) and the derivative of the inner function u, du/dx, then all we need to do is integrate f(u) with respect to u.
What this means is to treat the inner u as a variable like you would normally do for x.
Here is a version without all my scribbles on it.
Alternatively,
There is a strong implication that this is the same as setting the inner function as u and using integration by substitution. As it turns out, it is the same thing but much faster!
And we have named this result the Reverse Chain Rule, because we (STEM people) are just creative and original like that.
Now it’s your turn!
As a wise man once wrote:
“You are stranded on a desert island with a Swiss army knife, a boxed set of old Abba hits and a pair of Calvin Klein jeans. How would you integrate the following with respect to x?”
— Orlando Gough, British composer and author of ‘The Complete Advanced Level Mathematics’
Once you have tried them for yourself, I have been generous enough (just kidding, it is only the decent thing to do) to post the solutions:
I hope you enjoyed reading my first maths post. I have no idea when the next one will be as I write these only when I get flashes of revelation from the maths gods. If you would like to see more regular content, I always churn out a few physics stories every week. A clap and follow are much appreciated!
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