I know that when you're trying to integrate something, and by chance, appears somewhere something of the form :
$$ \sqrt {1 - x^2} ; \sqrt {1 + x^2}$$ or something similar, it is known to replace x by a trigonometric function, here
$$ \sin y \, ; \, \sinh^{-1} y $$
Inversely, if you're stuck with trigonometric function like in this case : $$ \frac{1}{\sin x}$$ it is good to use Tangent half-angle substitution, or also called Weierstrass substitution in order to create a rationnal function.
Finally (and that's the best), when you have any function and you see that the integral is even or odd, it is recommanded to use the substitution of either $x = \cos y$ or $x = \sin x$. I don't know how this law is called in English, but in french we call it "règle de Bioche".
My question is the following : Why is it always working? Why those earmarks implies that doing this substitution is going to work ?
I was believing that maybe rational and trigonometric functions have a link, that somehow, trigonometric function and roots are linked ?
Nevertheless, when I plot the graph, I don't see anything. Maybe someone has a better intuition and have an explanation of why those substitution works, and how is it visible.
If you need more example to understand what I mean, please take a look at this : https://www.youtube.com/watch?v=l1-xHNPMpB0 https://www.youtube.com/watch?v=ria48cFHkBw https://www.youtube.com/watch?v=TuEmoUU60FY https://www.youtube.com/watch?v=-rRaDRQ7ans
Moreover, another example is Euler Substitution, where does it comes from ??? it is so weird...