I was attempting a problem, when I was met with a confusion regarding the limits of integration. The problem was the following
$$ \int_{1}^{2} x^{2} \mathrm{d}(\ln x) $$
To clarify any confusion, $\ln x$ is inside the differential. My first intuition was to write $\mathrm{d}(\ln x)$ as $\frac{1}{x}\mathrm{d}x$, and proceed with the integral as usual to yield $\frac{3}{2}$ as the answer. However the answer given is $\frac{e^4 - e^2}{2}$.
I thought the limits of the function then would represent values of $\ln x$ from $1$ to $2$ and not the values of $x$ and that apparently yields the correct answer. But upon researching to confirm this, I found on Wikipedia Rienmann-Stieltjes Integral, where they imply
$$ \int_{a}^{b} \mathrm{f}(x) \mathrm{d}\mathrm{g}(x) = \int_{a}^{b} \mathrm{f}(x) \mathrm{g}^{\prime}(x)\mathrm{d}(x) $$
When $f(x)$ is bounded and $g(x)$ is increasing, continuous & differentiable in $[a, b]$. The same can be found out in Apostol's Mathematical Analysis Ch7 Sec1.
From this it appears my answer is correct. So is the book wrong and the limits do represent values of $x$ from $1$ to $2$ or is there something wrong? Thanks.
First of all, this notation is awful and is evidently prone to misunderstanding. Here, they mean that $$\int_{\ln(t) = 1}^{\ln(t)=2} t^2 d(\ln(t)) = \int_{e}^{e^2} t^2 \frac{1}{t} dt = \frac{e^4-e^2}{2}.$$ But this probably originated from the variable change $x = \ln(t)$ in the integral $$\int_1^2 (e^x)^2 dx,$$ which is obviously a stupid way to calculate this integral.
In summary, never do what they did here! Always be clear, so try to avoid notations like $d(g(x)),$ which is for physicists.