What does $\int d(\sin x)$ actually mean?I mean to say when we have $\int f(x)dx$,we interpret it as the values $f(x)$ s are being multiplied by tiny $dx$ and are summed together.But how do we interpret the above from this point of view.Because there is $d(\sin x)$,not $dx$ and we take $\sin x$ in the Y axis.But in integration we take dx to be in the X axis.
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$dx$ represents tiny change in $x$. Equivalently $d\sin x$ represents tiny change in $\sin x$. We generally do like this if $f(x)$ can be expressed as a function of $\sin(x)$. – Shubham Johri Nov 10 '20 at 14:28
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Duplicated: https://math.stackexchange.com/questions/295383/integration-of-a-function-with-respect-to-another-function – Clemens Bartholdy Nov 10 '20 at 16:03
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1Does this answer your question? Integration of a function with respect to another function. – Clemens Bartholdy Nov 10 '20 at 16:03
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as
$$\frac{d}{dx}\sin x=\cos x$$
$$d(\sin x)=\cos x~ dx$$
Thus
$$\int d(\sin x)=\int \cos x ~dx=\sin x+C$$
It is used when you do a variable's change.
Example
$$\int \sin^{10}(x) \cos(x) ~dx=\int \sin^{10}(x) d(\sin(x))=\frac{\sin^{11}(x)}{11}+C$$
K.defaoite
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tommik
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@tommik's answer is correct in that we do use this notation as shorthand in, for example, change of variables. However, I think you are looking for an intuition behind what the notation means. The intuition of taking values $f(x)$ and multiplying them by small $\Delta x$ and then summing them up is the Riemann Integral. When we start to have integrals like $\int d(f(x))$ we are moving into the realm of the Riemann-Stieltjes Integral.
The way that I like to think about this (which offers a reasonable generalization to integrating with respect to functions that are not continuous or random variables, as well as to measures) is that we are weighting our divisions not by their length, but by some arbitrary function. So in the example of $\int d(\sin x)$, we are integrating the (constant) function $1$, but instead of weighting the subdivisions of the domain by the length, we are weighting them by the sines of the endpoints (specifically the small change in sine on that interval that @ShubhamJohri mentions in a comment). When our integrator is differentiable, this is equivalent to integrating the derivative with respect to $x$, which helps justify the notational choices.
