The Substitution Rule - explaining dx

[jpanknin]

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

The original problem is [math]\int 2x \sqrt{1+x^2} \,dx[/math]
Then[imath]\ u[/imath] is set to [math]\ u = 1 + x^2[/math]
And finally[imath]\ du[/imath] is set to [math]\ du = 2x dx[/math]
My question is: when setting[imath]\ du = 2xdx[/imath], what does the[imath]\ dx[/imath] 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.

 

[Harry_the_cat]

\(\displaystyle du = 2x dx\) is another way of saying \(\displaystyle \frac{du}{dx} = 2x\).

It allows us to write\(\displaystyle \int 2x \sqrt{1+x^2} dx\) as \(\displaystyle \int\sqrt{u} du\).

 

[jpanknin]

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[imath]\ du[/imath] and the[imath]\ dx[/imath]? 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.

 

[Harry_the_cat]

\(\displaystyle du = 2x dx\) is another way of saying \(\displaystyle \frac{du}{dx} = 2x\).

It allows us to write\(\displaystyle \int 2x \sqrt{1+x^2} dx\) as \(\displaystyle \int\sqrt{u} du\).

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.

 

[Dr.Peterson]

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[imath]\ du[/imath] and the[imath]\ dx[/imath]? 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.