I'm learning Trigonometric Substitutions, they gave us the following example in the book:
I'm confused about how exactly we make the substitution $x= a\sin(\theta)$
In regular substitution we have to take something from the integrand and substitute it for u. Why is it in this case we can use $a\sin(\theta)$ when I do not see it in the integrand?
It doesn't really matter whether you 'see' anything in the integral (although sometimes picking your substitutions based on what you can see is useful). What you are doing, is defining a new variable in terms of the old one, and rewriting the expression you have entirely in terms of the new variable.
You can think of this example as defining $\theta$ by the formula:
$$\theta=\sin^{-1}\left(\frac{x}{a}\right)$$
from which it follows that $x=a\sin(\theta)$. Then to rewrite the expression in terms of $\theta$ you can simply replace all occurrences of $x$ with $a\sin\theta$.
This is analogous to beginning with an expression such as $x\sqrt{2x-1}$. Seeing the $2x-1$ inspires you to make the substitution
$$u=2x-1$$
(however you could've defined $u$ in this way, even if $2x-1$ didn't appear in the expression). It follows from this that $x=\frac{u+1}{2}$, so to write this expression in terms of $u$, you would replace all occurrences of $x$ with $\frac{u+1}{2}$. (Of course, you chose $u$ so that the expression under the square root would be equal to $u$, so you could instead replace the $2x-1$ immediately with $u$ -- you will get the same answer).
When I teach trigonometric substitution I emphasize that it is an implicit substitution. In contrast, the usual $u$ substitution is an explicit substitution. An implicit substitution introduces a new variable through some equation linking it to the given integration variables. In contrast, an explicit substitution, which you are more familiar with writes the new variable in terms of the given. In both cases, we have two or three things to do:
The details of 1 and 2 look different for implicit vs. explicit, but it's the same concept just like differentiation and implicit differentiation are really the same concept. Pragmatically, the trig. substitution is easier to find $dx$ in terms of the new measure $d\theta$ because you just take $x=a \sin \theta$ and find $dx = a \cos \theta d\theta$. So, how to change $dx$ to the corresponding expression with $\theta$. The answer to the preceding question is obvious; $dx = a \cos \theta d\theta$. Finally, how to change the integrand? Well, your post already shows how the Pythagorean identity for sine and cosine obliterate the root.
It is important to notice the new variable $\theta$ is the analog of usual $u$-substitution you've studied previously. By the way, there are also $u$-substitution problems where the $u$ is not in the original integrand, those are just unusual problems. For example, $\int \sec \theta d\theta = \int du/u$ for $u=\sec \theta+\tan \theta$. (see, no $u$ in the original problem)
