Is the following notation common in calculus books? $\int_{0}^{\frac{\pi}{2}} (1-\sin^2\theta)\,d(\sin\theta)$

Question

I am reading a calculus book.

$$ \int_0^{\pi/2} \cos^3\theta \,d\theta = \int_0^{\pi/2} (1-\sin^2\theta)\,d(\sin\theta) = \left[\sin\theta-\frac{1}{3}\sin^3\theta\right]_0^{\pi/2} = \frac{2}{3}. $$

Is the following notation common in calculus books?

$$\int_0^{\pi/2} (1-\sin^2\theta)\,d(\sin\theta)$$

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I think the author considers $\sin\theta$ as a variable.

So, I think the following notation is better:

$$\int_0^1 (1-\sin^2\theta)\,d(\sin\theta)$$

Is the above notation uncommon in calculus books?

Answer 1

American Calculus books usually use such notation in the context of Riemann-Stieltjes integration, but not for regular Riemann one.

I know the Russian analysis books use this notation very frequently. It's a nice shorthand for substitution.