ok..sorry for such an easy qeuestion but i have hit a complete mind blank...
what is the integral of
i know the answer it is....e^2x / 2
sorry but i have 2 exams and been studying for both of them forgetting one or the other...
It's amazing I ever got past pre-algebra.
Is there an official term for this? I've been using anti-substitute, because it makes me happier than "desubstitute" or "unsubstitute" or "remove the substitution" Although I have to agree with Krizalid that Soroban's "back u into x" is quite amusing. Although "Anti-substitute" has a nice fatal ring to it, which compliments that knot of dread in my stomach that comes from the scary way I integrate.
I got the impression the OP knew the answer because of this, but needed it to be shown with substitution for their test.
indeed!
pretty much any substitution problem can be done in this way. after all, integration by substitution was developed to undo the chain rule. (integration by parts matches the product rule etc...)
i assumed the poster wanted the standard/traditional way to integrate this...
hehe
Originally Posted by Krizalidit is not just cool and amazing, but it is the official term. the text books i have read use "back-substitute" to describe the process.Is there an official term for this? I've been using anti-substitute, because it makes me happier than "desubstitute" or "unsubstitute" or "remove the substitution" Although I have to agree with Krizalid that Soroban's "back u into x" is quite amusing.
ok, fair enough. keep using it. there's no problem.Originally Posted by angel.white
i found the term to be, well, cool, which is why i said anything
i did too.Originally Posted by angel.white
ah! the standard way to do this is to do integration by parts on the original, that is, , using and . so that would save you the trouble of your first substitution
the way, Krizalid, i believe, did it, was to, again, reverse the product rule
consider
we know,
now integrate both sides and solve for , and the result follows.
incidentally, the same reverse-product rule method could be used on , as opposed to integration by parts or tabular integration (if you're like Krizalid, i guess)
Yes, having (where is a positive integer) or the well known improper integral for a double integration trick works nice for both of them. (Also series and gamma function for the last one.)
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About without integration by parts, it's like Jhevon did: define contemplate its derivative and integrate. (Some integrals can be killed with this method.)