Calculus The Integral
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The Integral

 
 When we find the derivative of something like:
 

 F(x) = 3X5

 
 We use the formula:

 The derivative of AXn is (n x A)X(n-1)

 
 So using that on the F(X) = 3X 5, we get

 F'(x) = (5x3)X(5-1)

  F'(x) = 15X4

 
 Now we are going to reverse the process, and call it the integral.
 
 We will start with the derivative 
 and figure out the original function it came from.
 
 We will start with stuff like 15X 4
 and turn it into 3X 5.
 
 We just do the exact opposite process.
 Instead of multiplying, we divide.
 Instead of subtracting we add.
 
 So we use the formula:

The integral of AXn is  

A X(n+1)

n+1
 
 Since finding the integral is supposed to be the opposite
 of finding the derivative,
 you should be able to start with something, like maybe 4X 6,
 find the derivative of it, then find the integral of that,
 and be right back where you started at 4X 6.
 
 Let's see if it works:
 
 The derivative of 4X6 is:

 (6x4)X(6-1)

24X5

 The integral of 24X5 is:
24 X(5+1)

5 + 1

4X6

 
 IT WORKS!!!
 
 In general, it does work.
 There is, however, a case where it doesn't work.
 
 The derivative of 2X 4 + 5 is:

 (4x2)X(4-1) + 0

8X3

 
 The integral of 8X 3 is:
8 X(3+1)

3 + 1

 2X4

 
 WHAT HAPPENED TO THE " + 5" PART????
 
 It got lost in translation.
 Unfortunately, that happens to terms that are just numbers (called constants).
 
 The derivative of a constant is zero. Once it's gone, it's gone.
 If we had just found the derivative ourselves,
 we would know what constant had been there (if any).
 If somebody comes along later and just sees the derivative, they won't.
 
 Math people worried about that.
 A lot.
 Yes, they're weird.
 
 What they finally decided to do was this.
 If we find the derivative of something
 and later on somebody else uses it to find an integral,
 any constants that had been there would be lost.
 
 So when we do the integral we will add a "+ C" to the end to say:
 "Maybe there was a constant here that got lost along the way."
 
 So that means that if we want to be COMPLETELY accurate
 (and of course we do!),
 we would say:

The integral of AXn is  

A X(n+1) + C

n+1
 
 This kind of integral is called an "Indefinite Integral"
 It is indefinite, because we just don't know what that + C thing stood for,
 it could have been anything.
 
 If this is the Indefinite Integral, 
 does that mean that there is something called a Definite Integral?
 
 Yup! Sure does.
 And we'll get to that in a few pages.
 
 When we do derivatives, we have a bunch of different symbols that say
 "the derivative."
 
 Integrals only have one symbol.
 It is a big funny looking "S" that goes to the left of the terms.
 The integral notation also has a "d",
 followed by the independent variable at the end of the terms.
 
 So the code for:
 "The integral of AXn where X is the independent variable" is:

 

 So putting all this stuff together, we get:

 

 
 Well that's an impressive group of numbers, letters, and squiggles!
 Drop this puppy on your friends and watch what happens...
 But taking it piece by piece, it's no big deal.
 
 Let's do a few:

 

 
 When we found derivatives, and had something like:
 

 3X4 + 2X3 + X2 - 10X 

 
 We found the derivatives of each term separately.
 We do the same type of things with integrals.
 

 

 is the same as:

 

 
 Which works out to:

 

 Don't forget the "+ C" part!
 
 If we have a problem with a denominator, we can still sometimes find the integral,
 if we can simplify it.
 We can't easily find ...

 

 But we can change this to:

 

 Which simplifies to:

 

 And that we can work on a term at a time ...

 

 Which works out to:

 X3 - X2 - 3X - 4X-1 + C

 Don't forget the silly "+C" thing!!!
 
 If we have a coefficient that's too confusing to work with,
 we can just move it to the left of the big funny "S" and deal with it later.
 

 

 As always, the "+ C" comes along for the ride.
 
 A word about that dX thing
 Once upon a time, I said that the dX tells you that X is the variable.
 That's true, but it's not the whole story.
 When we find the derivative of something like (stuff) n,
 we do an outside/inside type derivative.
 
 The derivative of (stuff) n is:

 n(stuff)n-1 x (derivative of stuff)

 
 Whe we find the derivative of X n, we get nX n-1dX.
 The dX is the derivative of X.
 If the X is just an independent variable, which it usually is,
 then dX = 1.
 If dX is equal to 1 , we can multiply something by it
 and not change the value.
 
 Whenever we need to find the integral of some function of X,
 we need to have a dX to "put back in" when we do this reverse process.
 Since dX = 1, we can just tack it on anywhere we need it.
 
 This dX trivia probably seems pretty dumb and boring,
 but later on when the equations are more complex it will come in handy.
 
 Then you'll say:
 "Oh yeah, I remember reading about that,"
 "I didn't have a clue what that was all about then."
 
 But at least now you "know" about it.
 

   copyright 2005 Bruce Kirkpatrick

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