Use trigonometric substitution to evaluate
I see no reason not to follow their example.
Note when you differenciate you get the differencials,
But can you explain what those differencials mean!
They are not numbers!
Can you understand why it works? I cannot!
Thus, I use the strict form of the rule,
If has an antiderivative on some open interval and is differenciable on this interval then,
Where, is any function which satisfies the differencial equation throughout the interval.
If you can formally prove to me how these differencials work then I will use them, but I never understood them and willing to bet most people do not understand them either (if any at all).
Another thing, which makes me really angry is the symbol after the integral,I myself would rather drop it. In fact, my formal, differencial equations professor, does not write it. And my 12th Grade teacher also had a strong dislike to it, he used the composite function method (chain rule). Again I myself dislike it, greatly. I am sure if you open any analysis book they do not use this method, rather the correct one.
I believe these symbols are historical. They came from the time of Leibniz and remained until today. But the problem with Calculus (or "Method of Fluxions") in the 17 Century was that many mathemations disfavored it and opposed it (for example, Michelle' Rolle). They claimed it used "unsound reasoning", which in fact is true. Derivations using infitesimal arguments looked appealing but they were not based on mathematical derivations and thus were rejected by the mathematicians (but accepts by physcisits). Until the Cauchy and Weirestrauss became to build a foundation mathematicians did not approve Calculus.
I understand, topsquark, that this can produce the correct result but in strange cases this will probably not work. It is important for a mathematician to understand how something work rather than use it and know that it works. This is not alchemy.
I believe thee need a mathematicans apology.
Just giving you apologies in advance for my "informal" methods!