U-substitution notation in indefinite integrals

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?

• The answer is in the question: because it gives correct answers. It's easier to remember, but it is sloppy. I agree we shouldn't use it, but what can you do. – 5xum Feb 22 '16 at 10:31
• Here another point of view. – N74 Feb 22 '16 at 11:05