They were found either using trial and error or through proofs and are chosen based on if they can simplify the calculation. There are many videos on integration by parts and trigonometric substitution that will explain in detail why and when to choose which simplification. When you learn what the different formulas become when derived and integrated, you will start to become comfortable with their substitutions, it just requires some practice.
4:41 from here you can say that inner function is odd, and the int from -a to a of an odd function is 0
very nice
such an interesting integral haha...btw do you know any good books which can enhance your ingetration level (intermediate-advanced)?
Trig sub was always so strange for me... how to choose a correct trig function and why that is working...
They were found either using trial and error or through proofs and are chosen based on if they can simplify the calculation.
There are many videos on integration by parts and trigonometric substitution that will explain in detail why and when to choose which simplification.
When you learn what the different formulas become when derived and integrated, you will start to become comfortable with their substitutions, it just requires some practice.