# What does integrating a function $f(x)$ with respect to a function $g(x)$?

I encountered the following question in my book:

"Integrate $f(x)=\sqrt{1+x^2}$ with respect to $x^2$."

I am a bit confused about what this is supposed to mean.

In general, what does it mean to integrate a function $f(x)$ with respect to a function $g(x)$?

• My guess is the Riemann–Stieltjes integral. – Harald Hanche-Olsen Oct 25 '12 at 10:21
• Probably that book has said what it means at some previous place... – GEdgar Oct 25 '12 at 11:50
• Off-topic note. There are 51 users here with username "Chris" or "chris". – GEdgar Oct 25 '12 at 11:52

In my opinion you are working with Stieltjes integration. See here for a detailed introduction and examples:

http://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral

http://en.wikipedia.org/wiki/Lebesgue%E2%80%93Stieltjes_integration

Let $u=x^2$ now integrate $(1+u)^\frac{1}{2}du$,
after integrating sub back in $u=x^2$. thats it!
note: theres no need to find relation between $u=x^2$ as in $\frac{du}{dx}=2x$ shouldnt be substituted in.
• It's correct. Some differential calculus tells you that $d(x^2)=2xdx$. We can then substitute $u = x^2$, making $du = 2xdx$, and your integral results. – Lord_Farin Oct 25 '12 at 10:35