Integration of a function with respect to another function. What is the intuition/idea behind integration of a function with respect to another function? Say $$\int f(x)d(g(x)) \;\;\;\;\;?$$
or may be a more particular example $$\int x^2d(x^3)$$

My concern is not at the level of problem solving. To solve we could simply substitute $u=x^3$ and then $x^2=u^{2/3}$. My concern is rather about what meaning physical/geometrical does this impart?

ADDED if you ask what kind of meaning I seek, integration of a function w.r.t a variable gives the area under the curve and above x-axis.
There's no geometric interpretation of $\int f(x) \mbox{d}(g(x))$ as far as I'm concern. However, it does have a meaning. Say, you want to take the differentiation of $\sin(x)$ w.r.t x, and $\sin (x)$, they give 2 different results:


$\frac{\mbox{d} \sin x}{\mbox{d}x} = \cos x$
$\frac{\mbox{d} \sin x}{\mbox{d} \sin x} = 1$


Another example is to take the derivatives of $x^4$ w.r.t x, and $x^2$ respectively.


$\frac{\mbox{d} x^4}{\mbox{d}x} = 4x^3$
$\frac{\mbox{d} x^4}{\mbox{d} x^2} = \frac{\mbox{d} (x^2)^2}{\mbox{d} x^2} = 2x^2$


An antiderivative of $f$ w.r.t $x$ is some function $F$, such that $\dfrac{\mbox{d}F}{\mbox{d}x} = f$

So:


$\int \mbox{d}(x^2)$ is some function, such that its derivative w.r.t $x^2$ is 1, this family of function is, of course, $x^2 + C$.
$\int x^5\mbox{d}(x^5)$ is some function, such that its derivative w.r.t $x^5$ is $x^5$, this family of function is, of course, $\dfrac{x^{10}}{2} + C$, since: $\frac{1}{2}\dfrac{\mbox{d}(x^{10} + C)}{\mbox{d}(x^5)} = \frac{1}{2}\dfrac{\mbox{d}((x^5)^2)}{\mbox{d}(x^5)} + 0 = x^5$




When solving problems, you just need to remember that $\int f(x)\mbox{d}(g(x)) = \int f(x).g'(x) \mbox{d}x$, e.g, when you take some function out of d, you have to differentiate it, and vice versa, when you put some function into d, you'll have to integrate it, like this:


$\int x^5 \mbox{d}(x^2) = \int 2x^6 \mbox{d}(x) = \dfrac{2x^7}{7} + C$. (Take $x^2$ out of d, we differentiate it, and have $2x$)
$\int \sin^2 x \cos x \mbox{d}x = \int \sin^2 x \mbox{d}(\sin x) = \frac{\sin ^ 3 x}{3} + C$ (Put $\cos x$ into d, we have to integrate it to get $\sin x$).
This is an old question with a marked answer, but there is a better one.

This is how you find the area under the curve for a function with respect to another function given some hidden parameter. One such application is the receiver operating charateristic (ROC)
Receiver Operating Characteristic

In the ROC, we are interested in how the dependent variables $f(t)$ and $g(t)$ vary with $t$. We have the probability of false detection on the $x$-axis and the probability of detection on the $y$-axis. So, the integral of $f(t)$ with respect to $g(t)$ is an excellent measure of how likely we are to correctly identify the presence of some signal. There are likely other similar interpretations, but this is one I'm familiar with.