Question Number 91037 by M±th+et+s last updated on 27/Apr/20
$${hi}\:{every}\:{one} \\ $$$${what}\:{is}\:{the}\:{scientific}\:{reason}\:{for}\:{using} \\ $$$${trigonometric}\:{compensation}\: \\ $$$${in}\:{integration}? \\ $$$${and}\:{what}\:{is}\:{the}\:{rule}\:{that}\:{we}\:{rely}\:{on} \\ $$$${in}\:{other}\:{compensation}? \\ $$$$ \\ $$$$\sqrt{{a}^{\mathrm{2}} −{x}^{\mathrm{2}} }\rightarrow\rightarrow{x}={a}\:{sin}\left(\theta\right)\:{OR}\:{a}\:{cos}\left(\theta\right) \\ $$$$\sqrt{{a}^{\mathrm{2}} +{x}^{\mathrm{2}} }\rightarrow\rightarrow{x}={a}\:{tan}\left(\theta\right)\:{OR}\:{a}\:{cot}\left(\theta\right) \\ $$$$\sqrt{{x}^{\mathrm{2}} −{a}^{\mathrm{2}} }\rightarrow\rightarrow{x}={a}\:{sec}\left(\theta\right)\:{or}\:{a}\:{csc}\left(\theta\right) \\ $$