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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 $g(x)$?

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    Off-topic note. There are 51 users here with username "Chris" or "chris".2012-10-25

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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

http://ocw.nctu.edu.tw/upload/classbfs1209122139184046.pdf

http://www.math.mcgill.ca/labute/courses/255w03/L1.pdf

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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.

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    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.2012-10-25