(x+y)^2 is not equal to x^2+y^2 you have to remember this because  
(x+y)^2 is not equal to x^2+y^2 you have to remember this because  

otherwise the cat will be sad alright?
otherwise the cat will be sad alright?

okay to  expand X Plus y squared we will have to write this down twice so we have X Plus y times X Plus  Y and then we are going to use the foil method  
okay to  expand X Plus y squared we will have to write this down twice so we have X Plus y times X Plus  Y and then we are going to use the foil method  

that means we are going to do the first times  first x times x is x square and then the outer  
that means we are going to do the first times  first x times x is x square and then the outer  

x times y we are just going to write down plus x  y and then the inner y times x is the same as x  
x times y we are just going to write down plus x  y and then the inner y times x is the same as x  

times y so we'll just put on plus X Y again and  last times last y times plus y squared and then  
times y so we'll just put on plus X Y again and  last times last y times plus y squared and then  

we see that this then that can be combined it so  the answer is x squared plus 2 X Y plus y squared
we see that this then that can be combined it so  the answer is x squared plus 2 X Y plus y squared

okay to do X Plus y Square first we're going to  say this much right here is X and this much right  
okay to do X Plus y Square first we're going to  say this much right here is X and this much right  

here is y so the whole thing is X Plus Y and 2  square means that we are just going to get the  
here is y so the whole thing is X Plus Y and 2  square means that we are just going to get the  

area of a square so right here let's put down  X right here as well and then also white right  
area of a square so right here let's put down  X right here as well and then also white right  

here as well and then now let's just make  a big square like so aha really cool huh  
here as well and then now let's just make  a big square like so aha really cool huh  

and then we can see that this area is x times x  so that's x square and this area is x times Y and  
and then we can see that this area is x times x  so that's x square and this area is x times Y and  

then this area is x times Y and then lastly this  detail area is y times Y which is y squared so as  
then this area is x times Y and then lastly this  detail area is y times Y which is y squared so as  

you can see all together we have x square and then  we will have to add two of this so that's plus 2  
you can see all together we have x square and then  we will have to add two of this so that's plus 2  

x y and lastly we will have to add y squared and  that's the answer all right in order for us to do  
x y and lastly we will have to add y squared and  that's the answer all right in order for us to do  

this we will first need some definitions so let  me write them down right here let's define what  
this we will first need some definitions so let  me write them down right here let's define what  

x is first and of course in calculus we'll try  to Define this as an integral let's define X to  
x is first and of course in calculus we'll try  to Define this as an integral let's define X to  

be the integral going from 0 to X of 1 DT next we  need to have x to the second power and again we'll  
be the integral going from 0 to X of 1 DT next we  need to have x to the second power and again we'll  

Define this as an integral as well the integral  going from 0 to X of 2T DT now we can only use  
Define this as an integral as well the integral  going from 0 to X of 2T DT now we can only use  

these two definitions and we see that here we have  X Plus y Square so we will use this definition  
these two definitions and we see that here we have  X Plus y Square so we will use this definition  

this means we have this being the integral  going from 0 to X Plus Y and then we have 2T DT  
this means we have this being the integral  going from 0 to X Plus Y and then we have 2T DT  

but here we can integrate and plug in because  that defeats the purpose we can only use the  
but here we can integrate and plug in because  that defeats the purpose we can only use the  

properties of integrals to take care of this how  do well this integral goes from 0 to X Plus y  
properties of integrals to take care of this how  do well this integral goes from 0 to X Plus y  

That's separate it let's talk about this as the  integral going from 0 to X first and then 2T DT  
That's separate it let's talk about this as the  integral going from 0 to X first and then 2T DT  

and then we will have to add pick this up from X  and then goes to X Plus Y and then we have 2T DT  
and then we will have to add pick this up from X  and then goes to X Plus Y and then we have 2T DT  

this is perfect because this right here  by definition is x square so that's great  
this is perfect because this right here  by definition is x square so that's great  

but now what is this though as we can see  this intake will start with X we don't want  
but now what is this though as we can see  this intake will start with X we don't want  

that because in water for us to use these two  definitions we will have to make sure the intake  
that because in water for us to use these two  definitions we will have to make sure the intake  

will start with zero so right here let's just  do some substitution and keep in mind we are in  
will start with zero so right here let's just  do some substitution and keep in mind we are in  

the T world and we have ax right here so let's  just go ahead and have T minus X so we can get  
the T world and we have ax right here so let's  just go ahead and have T minus X so we can get  

the zero and let's call this to be a new variable  let's say U and now let's move this to the other  
the zero and let's call this to be a new variable  let's say U and now let's move this to the other  

side so we get T equals U plus X differentiate  this we will get DT equals du the derivative  
side so we get T equals U plus X differentiate  this we will get DT equals du the derivative  

of x in here will be 0 because X is like a  constant here now we can take this and then  
of x in here will be 0 because X is like a  constant here now we can take this and then  

fix it and see what we get keep in mind though  this is T is equal to X we will have to put it  
fix it and see what we get keep in mind though  this is T is equal to X we will have to put it  

here so that means U will be x minus X which  is zero so we will have U goes to be 0 first  
here so that means U will be x minus X which  is zero so we will have U goes to be 0 first  

and then we will have to put X Plus 1 here  the X will cancel so we'll just have the Y  
and then we will have to put X Plus 1 here  the X will cancel so we'll just have the Y  

and then we have the two and the t is U plus  X and the DT is the same as the U very nice  
and then we have the two and the t is U plus  X and the DT is the same as the U very nice  

now we're just going to take off this this way  I'm going to distribute this so we have 2 U and  
now we're just going to take off this this way  I'm going to distribute this so we have 2 U and  

then that will be the integral going from 0 to  Y and then 2u and let's close that we have the  
then that will be the integral going from 0 to  Y and then 2u and let's close that we have the  

view here and then we are going to add okay  2 times x but they are both constants in the  
view here and then we are going to add okay  2 times x but they are both constants in the  

U world so you can take it on the outside right  here and then look at the integral going from 0  
U world so you can take it on the outside right  here and then look at the integral going from 0  

to Y and then this right here is just going to be  the U inside now what's this well this right here  
to Y and then this right here is just going to be  the U inside now what's this well this right here  

is pretty much similar to this right which is  something square and here we have the Y squared  
is pretty much similar to this right which is  something square and here we have the Y squared  

then huh so this part gives us y squared and  then the 2x is just 2X and now what's this part  
then huh so this part gives us y squared and  then the 2x is just 2X and now what's this part  

well this right here is like we have a one so  we are talking about this right here and when  
well this right here is like we have a one so  we are talking about this right here and when  

we have the X here that will give us X but here  we have the Y so this part will give us the Y  
we have the X here that will give us X but here  we have the Y so this part will give us the Y  

ladies and gentlemen this integral is precise  d y square plus 2 X Y again we have the  
ladies and gentlemen this integral is precise  d y square plus 2 X Y again we have the  

correct answer so I'm pretty sure all of your  t-shirts from pre-algebra to calculus do you  
correct answer so I'm pretty sure all of your  t-shirts from pre-algebra to calculus do you  

have told you that X Plus y squared is equal  to x squared plus 2xy plus y squared but now  
have told you that X Plus y squared is equal  to x squared plus 2xy plus y squared but now  

I wanted to tell you this right here might not  be true because X and Y might not commute so  
I wanted to tell you this right here might not  be true because X and Y might not commute so  

the best answer is X Plus y squared is equal  to x squared plus X Y plus Y X Plus y squared  
the best answer is X Plus y squared is equal  to x squared plus X Y plus Y X Plus y squared  

welcome to abstract algebra hey I hope you liked  the video and now I want to take some time to  
welcome to abstract algebra hey I hope you liked  the video and now I want to take some time to  

tell you guys about our sponsor today brilliant  they are a fun educational website and they have  
tell you guys about our sponsor today brilliant  they are a fun educational website and they have  

a big focus on Interactive Learning go ahead and  check out the pre-tocus course because this course  
a big focus on Interactive Learning go ahead and  check out the pre-tocus course because this course  

will enhance your understanding of the concepts of  exponential functions logarithms conceptions and  
will enhance your understanding of the concepts of  exponential functions logarithms conceptions and  

parametric equations one thing I really enjoy is  that they provide interactive lessons so that you  
parametric equations one thing I really enjoy is  that they provide interactive lessons so that you  

can easily see how the function has changed when  the numbers suggested I'd really like them because  
can easily see how the function has changed when  the numbers suggested I'd really like them because  

they also believe that the best way to learn is  by actually doing it rather than just memorizing  
they also believe that the best way to learn is  by actually doing it rather than just memorizing  

facts for exams you can just go ahead and pick a  course that you like and get started so go ahead  
facts for exams you can just go ahead and pick a  course that you like and get started so go ahead  

and check them out use the link in the description  brilliant.org black and red pen because this way  
and check them out use the link in the description  brilliant.org black and red pen because this way  

you can get 20 off discount and thanks so  much to brilliant for sponsoring this video
you can get 20 off discount and thanks so  much to brilliant for sponsoring this video