Hey guys welcome to a year 12 maths B tutorial in order to understand anything
about what I'm about to show you you need to have done the year 11 calculus
tutorials so make sure you have done that before continuing okay so we're
going to talk about anti-differentiation now what is anti-differentiation well if
you remember how to differentiate so if I give you a question like this
differentiate y equals 3x squared and you differentiate it and you can get dy
over DX equals 6x ah this is called differentiation right we have a function
which is y equals 3x squared and we differentiate it to give us the gradient
function which is dy over dx equals 6x now anti-differentiation refers to the
opposite of differentiation so I could give you the equation dy over dx equals
6x and I ask you what y is all right so that is the process of
antidifferentiation so how do you figure out if i give you an differentiated
equation how do you go back and find out what y equals well here is a general
formula that can help you find it so the formula is if dy over dx is equal to a
times x to the power of n then y is equal to ax to the power of
n plus one over n plus one plus C now note that this only works for n does
not equal negative one if n equals negative one then you need to use
another method to anti-differentiate it so let's try using this formula so now
we have dy over dx equals 6x and we want to know what Y is equal to so applying
the formula a is equal to 6 and we have X n is equal to 1 so it's 1 plus 1
over n is 1 so it's 1 plus 1 and plus C because we don't know what C is
so simplifying that we have 6x to the power of 2 over 2 plus C and simplifying
again we have y equals 6 divided by 2 is 3 x squared plus C which matches our
differentiation equation that we've just done earlier Y is equal to 3x squared dy
over dx is equal to 6x so this process of going from dy over dx to Y we call
that solving a differential equation in other words
anti-differentiate
now let's talk about this symbol here c what is this symbol doing why do we
need c because we know that 3x squared differentiates into 6x so why do
we need a constant c on the end well that is because if Y is equal to 3x
squared plus any number here that is just a number on its own so you're not
allowed to attach an X to the number so let's say I make up a number let's say 7
this will also differentiate into 6x so basically C represents any constant that
you can think of will be allowed in the Y function that will still give you a dy
over dx is equal to 6x as the differentiated equation so for example
if y is equal to 3x squared plus 15 y dash is still 6x so that's why we put a
symbol C there to represent that any constant at the back of the Y equation
will differentiate into this equation now the last thing that I'll talk to you
guys about is the symbol for antidifferentiation so if you see this
symbol if you see that symbol there and you see 6x
dx this just means find the antiderivative of 6x
all right so the answer is three x squared plus C right so 6x are three x
squared plus C is the antiderivative of 6x
okay so I've just written up some practice questions for you to have a go
at so try them out come back when you're done okay so let's
have a look at how to do this well following the formula I should probably
write down the formula first so you guys know uh how do you use it so I have dy
over dx equals ax N and Y is equal to ax n plus one over n plus one plus C so
therefore applying that formula I have Y is equal to four X now notice how I have
X to the power of one here so it's going to be four X to the power of 2 over 2
and then minus three that can also be anti differentiated so I have minus
three now remember X to the power of zero is equal to one so I'm just going
to add a little X to the power of zero there so if I anti-differentiate minus
three X to the power of zero I get minus three X to the power of 1 over 1 because
n here is equal to 0 0 plus 1 is 1 so therefore I also need to add another
plus C in the end so the answer is y equals 2x squared minus 3x plus C
so that's the antiderivative of dy over dx equals 4x minus 3 now if you
want to test whether you're correct you can simply differentiate this
function again so dy over dx so this is a little test dy over dx is equal to 4x
minus 3 so it matches what we had before ok let's try this one so y is equal to 3
over 4 X to the power of 2 over 2 plus C so you might think hang on this is a bit
confusing uh what do I do with this fraction within a fraction
well divided by 2 if you remember just means ah times by 1/2 all right plus C
now so what that means is you can just put 3x squared at the top because it's 3
over 4 times x squared so times x squared x squared goes to the top and on
the bottom you have 4 times 2 which is 8 so I have plus C again right now if you
want to test whether you have this correct you can differentiate it again
so dy over dx equals 2 times 3 over 8 so that's 6 over 8 X and 6 over 8 can be
simplified into 3 over 4 X so that is definitely the answer so take your
answer if you have it correct and the third one find the antiderivative of 4
x squared plus 3x so same thing 4 X cubed over
three plus three x squared over two plus C and that's it okay thanks for watching
guys see you next time.