



When we
find the derivative of something like: 





F(x) = 3X^{5}






We use
the formula: 


The
derivative of AX^{n} is (n x A)X^{(n1)} 





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


F'(x)
= (5x3)X^{(51)}



F'(x)
= 15X^{4}






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 AX^{n} 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 4X^{6} is: 


(6x4)X^{(61)}



24X^{5} 


The
integral of 24X^{5} is: 





4X^{6} 





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^{(41)}
+ 0



8X^{3} 





The
integral of 8X ^{3} is: 





2X^{4}






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 AX^{n} 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 AX^{n} 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: 





3X^{4}
+ 2X^{3} + X^{2}  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: 


X^{3}
 X^{2}  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)^{n1}
x
(derivative of stuff)






Whe we
find the derivative of X ^{n}, we get nX^{ n1}dX. 


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 
