Fun Stuff > CHATTER

Can't Think of a Breaking Bad Pun For the Title: Let's Do Some Math!

<< < (24/35) > >>

ankhtahr:
You won't believe how grateful we are for you Skewbrow. Without you we would have been doomed. With you, we've been a bit less doomed. It still took us a long time, but otherwise we wouldn't have figured out such nice solutions at all. My whole exercise-sheet-group has worked on this, and we wouldn't have made it without you.

Thank you.

Loki:
This is not under time pressure, as it was for an assignment which has already been turned in, but:

Let's say you integrate a function over the reals, for example f(x)=x*exp(-x). The way I learned it, if you integrate a product of functions, you can call one function u(x) and the other v'(x) and then do integration/derivation voodoo with them, which I believe is the inverse chain rule method.

Now, my question: How do I decide which of the two functions should be u and which should be v'? Is there a rule of thumb?

snalin:
Try one assignment, and if that's too hard, try the other. That was always my way of figuring it out.

As the ordering of the factors are irrelevant, which one you choose for u and v' is up to you - the result must always be the same.

Carl-E:
Actually, thereis a rule of thumb, and a mnemonic for it. 

The rule is that you want u to be something that gets simpler with derivation, and v' should be something that doesn't get much more difficult under integration. 

So, for example, if there's a logarithm, it's automatically u, because it can't be integrated without this trick anyway. 

So the rule of thumb; 

Choice of u should be

L - logarithmic terms (derivatives are negative powers of x, much simpler)
I - inverse trig terms (simpler derivatives, rational expressions, some with roots)
A - algebraic terms (polynomials, actually - the powers all get knocked down by one each time this is done)
T - trig terms (rarely simpler, but at least you're just getting other trig terms)
E - exponentials (no simpler, but no more complex when integrated, either, which makes it a good choice for v')

LIATE's not too hard to remember - it's almost a word.  If you have any two of these, use the first one that appears in the list should be used as u, the other one as v' and you'll usually get something simpler.  For your example,

u = x --> du = dx
v' = e^(-x) --> v = -e^(-x)

so int[x*e^(-x) dx] = (x)(-e^(-x)) - int[-e^(-x) dx] = -xe^(-x) - e^(-x) + C

BTW, this technique of reversing the terms in the product rule to get something easier is usually called "Integration by Parts", at least here in the US. 

And, lastly, this;

Schmee:

--- Quote from: Carl-E on 10 Jan 2014, 03:00 ---

--- End quote ---

Alt-text: "Eventually, a UN is set up. And then a lone rebel runs along the line of flags at the front, runs back to his base, and gets a kajillion points."

Navigation

[0] Message Index

[#] Next page

[*] Previous page