Calculus The Integral of 1 over X
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The Oddball Function
The Integral of 1 over X

 

 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

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