A formulation of mine, the trigonometric rule collapser set, is not to be confused for u-substitution. (See why)

In the sequence: x = sin t, where dx = cost dt, and 1 − x² = 1 − sin²t = cos²t ....(from problem: ∫ √1- x²)

..the novel formulation dx | dt · dx occurs such that the default way of working trigonometric equations is compressed, permitting the reduction of the cardinality of steps normally employable.

How does my trig rule collapser set collapse routines?

In the video above, the instructor solves ∫ √1- x² dx in the default way.

In the video while evaluating ∫ √1- x² dx, in some preliminary steps, the instructor writes ∫ (√1-sin²θ/(√cos²θ) ) · cosθdθ, then obtains cosθ from (√1-sin²θ/(√cos²θ) ).

Using my “trig collapser” routine, these steps (which may be more steps for other problems) are unnecessary, because applying my trig collapser set’s novel form dx | dθ · dx, one can just go right ahead and evaluate ∫cosθ·cosθdθ.