okay as you can see we have a rational function right here of course to integrate this let's try
to use partial fraction and of course we're to make sure that the degree on top is lower than
the degree on the bottom it is so we can call him proceed first this denominator we have to
factor that and to do so we're four terms so just at the cool algebra days will do this by
grouping so I would look at this 2 and this 2 and we can factor out x squared first and then we get
X minus 3 and then from here we can pull out pull out the 4 so we have X minus 3 like this and then
of course this and our common factor we can take that out as well so we have X minus 3 and then we
can just have x squared plus 4 like this all right so that's pretty much it for the factory and now
let me do the partial fraction on the side for you guys and yes x squared plus 4 we can i factor it
anymore you have to deal with this right here now let's talk about a set up you have two different
factors this right here linear this right here and we eat reducible quadratic so the set up is going
to be let's write this down first X minus 3 and because this is linear on the top it has to be one
degree lower which is just a constant which is put on a and then for this one x squared plus 4 well
as you can see this is quadratic in the situation on the top it has to be a linear namely BX plus C
it has to be one degree less than the bottom like this and for this one we can do copper up to take
up one that only named D a because I can go here and cover up X minus three and then I can make X
minus three things you roll by plugging Street into X so I already stomach here for you guys a
is equal to when you plug in 3 into the rest of the X right there so we end up three times three
squared and then minus 3 times 3 and then plus 8 all over this is out already putting strings
here so we get three square plus four like this all right now let's see if we can do this or not
hang your head let's see this is nine times I was 27 27 minus nine years 18 plus 8 here's 26
over this right here is 9 plus 3 is 13 so oh hey this is just nice number 2 that's good yep two
right here and now unfortunately we cannot solve for P or C by using a cover-up you cannot make X
square plus 4 equal to 0 with strong numbers and also you just don't know what's the cover-up so
to figure out B and C so just show you guys the traditional to do it will you'll come up with a
system of equations right so what we will do is I'll look at this and then multiply everybody by
X minus 3 times X square plus 4 and I'll see how this goes and if you really want you can plug in
the X equal to 0 X equal 1 because you haven't used them yet you can put into all the X and
in general equations you can do that all right but I think I will just show you guys this way
so when you do that this times that you just have this part left which is 3x squared minus 3x Plus
8 and you do this times that don't forget a is 2 already so you have to this times that the X
minus 3 cancel each other out so you have just 2 times this which is not that bad oh and then this
times that of course this and that cancel you have this right here which you don't know at the moment
I will just write it down B plus D in apprentices right here and you have to multiply by X minus 3
so you have to come up with system of equations if you don't want to plugging X values well I
think this right here it's study easier because you have two unknowns this right here taking of
that pretty easy just just to lay out where this is just algebra ok anyway here we go on left-hand
side we have a quadratic on the right hand side we don't know what he is where the clink sings up to
do so just multiply this out this is 2x squared plus 8 and then right here of course you do this
times this which is B X square this times this is minus 3 B X and then this times this is plus C X
and then this times this is negative 3 C here is the deal you are going to collect like terms as
you can see you have 2x squared plus BX square you can factor out the x squared and you end up
with two plus B and put the X square here next collect the linear term namely the negatives BX
plus CX and I will just put it as plus parentheses with negative 3b inside and then plus C and then
multiply by X on the outside lastly this is the Turner has no X likewise this so this is
the constant terms together you just say you can add 8 minus 3c like that and then if you look at
the left-hand side you have this is equal to this and here is the deal you're just going to equate
coefficients and this is how we can do it well the easier one is look on the left hand side we have 3
in front of the x squared that means this right here should also end out to be 3 isn't it yes
so this is very nice I can just put down we must have 2 plus B equals 3 and from here you can see
that B is equal to 1 very nice and you can also say negative 3 B plus C has to be negative 3 but
you don't have to because you can also use this if you like in fact they happen to be equivalent
if you put in 1 here you can negative 3 plus C does not be negative you end up the same as okay
by the way I will just do this right here for you guys this is equal to that which found here
B minus 3 C equals 8 this tells us that subtract 8 on both sides you get negative 3 C C is equal
to 0 this is very nice and again when you put in 1 here you can negative 3 B which is negative 3
C has to be 0 so just use two out of these three coefficients that will do the work you can use the
last one to double check but you don't have to now we'll come back here and then do the integration
here's the calculus step right ok first we have 2 over X minus 3 let me just write it down right
here for you guys and then next B right here is 1 so we know this right here is 1 and C is equal to
0 so you don't have this anymore you just have x over x squared plus 4 so I can just put this
down right here x over x squared plus 4 and in fact this is very nice ok integrate this
usually it's not a calculus that people hate it's the algebra hat like this that people don't like
you can kick calculus in its head okay seriously integrating this you get to natural log absolute
value of X minus 3 integrate this how do you do it yes you do u substitution I'll just read this
down here for you guys on the side okay thank you equal to x squared plus 4 and then you do
all that stuff ok I'll eat at you you add 1/2 also natural log in this case and you put it in
um to here X square plus 4 this is always positive you just need parentheses this right here is it a
plus C and you are done ok again work out a use substitution on your own this right here is it
hopefully this helps and let me know if you guys have any questions and I'll see you guys that's it