Posted by Soroban on September 04, 2002 at 23:07:42:
In Reply to: thank you posted by T.Gracken on September 04, 2002 at 19:56:01:
: : : How do I integrate this?
: : : dx/(a+Sqrt(x))
: : : Can someone explain the technique please?
: : : I don't need the actual answer, just help with the approach.
: : : I can do dx/Sqrt(x) fine (2Sqrt(x)) but this seems to be a beast of another nature...
: : First of all, T.Gracken's approach is correct -- and very courageous!
: : I assumed he'd already tried Substitution and didn't try it until just now.
: : And when I did -- WOW!
: : Let sqrt(x) = u
: : Then: x = u^2
: : And: dx = 2u du
: : Substitute: INT 2u/(u+a) du
: : Divide: 2 INT [1 - a/(u+a)] du
: : There!
: Thank you!
: I sometimes get tunnel vision and just go with it. For some reason I thought a basic substitution would have already been attempted and did not even consider a simple substitution (and manipulating the fraction by "adding zero to the numerator").
One of my favorite tricks is "adding zero",
'though I've never heard that phrase before.
I like it! (Funny that it didn't occure me -
I've been "multiplying by one" for years.)
: I should be severly whipped for all that extra crap I did.
Naw - I think the overheard snickering (whether
real or imagined) is punishment enough.
: ...I think I've been teaching too many developmental classes of late.
I know THAT feeling, too!
A long story - maybe you'll feel better, TG.
A colleague (working on his MS in math) asked for
help with a Diff.Equation. He had tried various
techinques: Homogenous, Cauchy, Integrating
Factors, Method of Operators, etc.
I worked on it for an hour and finally consulted
a reference book. There I found an obscure
theorem which solved the DE. I proudly showed
John my work; he thanked me and left for his
evening DE course.
Some time later, I looked again and saw that the
very complicated solution simplified drastically
- SO drastically that I was sure I had overlooked
something. I looked again...
Oh, no! It was VARIABLES SEPARABLE!
Talk about "tunnel vision"!
Luckily, I was able to phone John and tell him -
before he showed our intricate (and egregiously
stupid) solution to anyone.
That was twenty years ago.
I'm recovering nicely now, thank you for asking.
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