I originally posted this as a second part to a messy slew of questions I had on some lecture notes I had, so I decided to take the suggestion of some users and post this question individually.
I have a question I find I can't properly explain. In any integration by trig substitution problem you have to state something along the lines of this, for example:
$x = sin\ u$
What is this insinuating? Why am I allowed to even do this without it changing the problem entirely in the same way that this substitution is invalid?:
- $x + 4 = 10$
Let $x = sin\ y$
Therefore $sin\ y = 6$
Why is the above not okay other than the obvious that $sin\ y = 6$ is impossible yet it's okay in integration by trig substitution? Is that because of our changing the differential variable?
In cases where the substitution would require you to be able to say that sometimes $\sin y = 6,$ the substitution is not OK.
For example, you wouldn't make the substitution $x = \sin y$ in order to integrate $\int x^2 \,dx,$ partly because you already know how to do that integral without the substitution, but also because $x = \sin y$ would give you an integral that is valid only for $-1\leq x \leq 1,$ whereas the original integral is valid for all $x.$
On the other hand, if you have to integrate $$ \int \sqrt{1 - x^2} \,dx,$$ then the integral is already valid only for $-1\leq x \leq 1,$ because for $\lvert x\rvert > 1$ we would have $1 - x^2 < 0,$ and the square root of a negative number takes us outside the domain of real-number integration. So in that case you don't lose anything by substituting $x = \sin y.$