The substitution rule for indefinite integrals says that
$$\int f(g(x))g'(x)dx = \int f(u)du$$
where $u=g(x)$.
Probably the most typical example is $\int \sin x \cos x dx$.
Let $u=\sin x$. Then $du = \cos x dx$ and the integral becomes $\int udu$, because we changed the $\cos x dx$ into $du$.
Is this notation valid? I believe it might give a false impression that $\sin x \cos x dx$ is a multiplication of $\sin x \cos x$ by $dx$. There's no multiplication. $dx$ is there only to indicate the variable of integration.
While it gives correct answers, it's quite misleading. Then why is it used?
