mjhjmtu

If we consider: $int_a^bxdx$, where $a=sinx$, $b=cosx$.

How can we evaluate this particular integral, if $a$ and $b$ are both functions of $x$, which is the variable with respect to which we are integrating?

The variable inside the integral is a "dummy" in that it could be replaced by any other symbol. I think you could interpret $$int_sin x^cos x x , dx$$ as a sloppy way of writing, but having the same meaning as, $$int_sin x^cos x y , dy.$$

Here is another interpretation.

Formally if $bgeq a$ then $int_a^bxdx$ is a notation for $intmathbf1_(a,b](x)xdx$.

Applying that here leads to: $$int_sin x^cos xxdx=intmathbf1_(sin x,cos x](x)xdx$$

I do not dare to say that this is the correct way of interpreting, but it illustrates at least that your question is a good question.

Personally I go for the interpretation of Umberto.

Alternatively to Umberto P. you could also view it as:

$int_A f(x) dx$ where $A$ is a set, for example an area or volume, that is further distorted by your function $f(x)$.

In your case you would simply plug in your constraints: $A = sin(x) le a le cos(x) $ and for $f(x)$ you have of course: $f(x) = x$

Doesn't matter. Just your final solution will also be a function of x.

Proceed the usual way: find the indefinite integral $$int xdx=fracx^22$$
put limits: $a=sin(x)$ and $b=cos(x)$
$$=fraccos^2x2-fracsin^2x2$$
$$=fraccos(2x)2$$