With the new courses yet to start, hopefully providing fresh material for new posts, I have been spending time going through some excercises from new textbooks.
As integrals have always been my favourite part of calculus, I decided to take down this solution, because it just looks nice. It also illustrates the principle: don't make a substitution, until it becomes obvious.
MathJax failure: TeX parse error: Extra close brace or missing open brace
Since we have and , so we need to choose negative sign when taking square root of the quadratic term.
It's now that the substitute becomes an obvious choice.
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OBVIUOS??New comment
well, it's quite depressing when a textbook solution suggests a substitution "out of the blue". however, in fact the problem may often be cast into a form when change of variable is just "wrapping" of a frequently occurring expression and giving it a placeholder name. That's what I'm trying to achieve throughout my studies. Almost all "special functions" were introduced in a similar fashion.
On a broader scale I assume that different wrappings may lead to different pictures of the world...