Just trig? Heh. Something prompts me to comment beyond simply noting that algebra, trig, precalculus comprise the mechanics that drive the calculus machine. :cool:
When 'solution process' refers to a specific algorithm itself (eg: listing of integration steps), then I agree. We don't always include scratch-paper steps; some people reach a point where they don't even need paper.
Were 'solution process' to denote the overall process of solving an exercise or grasping a big picture, then I would say that modeling with a right triangle (on scratch paper or not) is a part of the process. It sure is for me because I tend to loose instant recall of things, especially things that I can derive. The Pythagorean identities you mention are a great way to help recognize which trig substitution is needed for certain integrands. A right triangle model can assist in correctly recalling these identities.
As an aside, I remember being away from trig for months at a time, from 9th grade through college, and beyond. Trig would suddenly reappear (sometimes like a locomotive on a freight train), and I regularly avoided panic by picturing x- and y-values of points on the unit circle, while recalling sine for vertical sides and cosine for horizontal sides of reference triangles, all to help me not mix up numerical trig values for the special angles. Or, simply using the right-triangle trig definitions of sine and cosine to label a trianglular model or to derive a tangent expression. In these examples, assigning trig expressions to the legs of a right triangle was part of my 'solution process', in a variety of different courses. Long story short, the more ways a student has to visualize models and interpret patterns, the more effective their solution processes become.
I like your explanation of 'trig substitution' as one of the techniques used to change a difficult integral into a form that you know how to work with. However, the example above seems to need some proofreading. Terminology technicalities aside (eg: notations like dx and dθ are not part of the integrand), there seem to be sign, substitution, or maybe typo errors, in the explanation for integrating sqrt(1-x^2) with respect to x. :cool: