The Substitution Rule - explaining dx

jpanknin

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Can someone explain the meaning/use/importance of dx in the following substitution?

The original problem is 2x1+x2dx\int 2x \sqrt{1+x^2} \,dx
Then u\ u is set to  u=1+x2\ u = 1 + x^2
And finally du\ du is set to  du=2xdx\ du = 2x dx
My question is: when setting du=2xdx\ du = 2xdx, what does the dx\ dx mean and what role does it play in the solution? I can use it to solve these problems because that's what the book says, but I don't fully understand the meaning/role of it.
 
du=2xdx\displaystyle du = 2x dx is another way of saying dudx=2x\displaystyle \frac{du}{dx} = 2x.

It allows us to write2x1+x2dx\displaystyle \int 2x \sqrt{1+x^2} dx as udu\displaystyle \int\sqrt{u} du.
 
Thanks @Harry_the_cat. I can use it mechanically to solve problems by following the formula in the book, but I don't UNDERSTAND what is happening and why it's happening. What happens to the du\ du and the dx\ dx? What do they mean and how why are they important? I'm really trying to UNDERSTAND what's happening here rather than just using a formula.
 
du=2xdx\displaystyle du = 2x dx is another way of saying dudx=2x\displaystyle \frac{du}{dx} = 2x.

It allows us to write2x1+x2dx\displaystyle \int 2x \sqrt{1+x^2} dx as udu\displaystyle \int\sqrt{u} du.
After an integral sign the dx tells us that we are integrating with respect to x.
du tells us we are integrating with respect to u. This is important because after we do the correct substitution, we can integrate w.r.t u.
I'm not sure if that answers your question though.
 
Thanks @Harry_the_cat. I can use it mechanically to solve problems by following the formula in the book, but I don't UNDERSTAND what is happening and why it's happening. What happens to the du\ du and the dx\ dx? What do they mean and how why are they important? I'm really trying to UNDERSTAND what's happening here rather than just using a formula.
What's happening here ultimately is the chain rule in reverse. The differentials provide a convenient way to keep track of things.

For a deeper explanation, see this page on my site.
 
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