



There
is one function where our integral formula does not work.



If we
have the equation: 





Then
applying the rule would give us: 





WOW! Try
to make sense out of that! 


On second
thought, don't. 





The real
answer is: 








OK,
What's an "ln" ??? 


"ln"
Lower Case "LN" is actually a special type of logarithm, 


with a base of about 2.71828. 


The
actual base number is a special irrational number. 


It is so
special, that it gets it's own letter for a name: "e" 


This
"e" thing has all kinds of applications, and we'll flog
them to pieces in time. 


For now,
just remember that: 








Why do
we need the absolute value sign??? 


The way logarithms
work, 


"lnX" means the exponent that 2.71828 needs to be
raised to, to get X. 


No matter
what power you raise 2.71828 to, you always get a positive number, 


so we use
the absolute value sign to make sure we know 


to have a positive
value for X. 





OK, Fine.
But what if we need the integral of some complex function like: 





F(x) = (3X
 5)^{1}



What
then??? 





Well then
we're back to the old outside/inside thing. 


We need
to have the derivative of the inside part sitting there at the
start. 


We have 


(3X
 5)^{1} which is the same as 
1 

3X
 5 






And we
need: 


(3X  5)^{1
}3dx which is the same as 
3dx 

3X
 5 






dx is
just 1 so that's no problem, 


but we need our little change the du
constant trick again... 


So: 








The
general form of this "ln" thing is: 








By the
way, even though we're not really looking for derivatives right now 


we've
stumbled on the fact that the derivative of lnX is 1/X. 


Hey, take
what you can get! 





Another
Example: 








copyright 2005 Bruce Kirkpatrick 
