All language subtitles for [English (auto-generated)] Integration by Substitution (2 of 4_ Doing the same Question with a simpl

so

i just said to you right this is this is

one way you could do it and i've picked

out a simple example where most of you

can like the numbers are not too

complicated you're like oh i can kind of

see that right and it only takes a

simple number to pull out the front okay

this is not the only way and sort of the

fact that we can do this sort of rides a

little bit on the fact that the numbers

are so easy if the numbers were a bit

more messy or if you didn't have

something as straightforward here you

can have another whole function there

this kind of technique becomes less and

less useful because it only works in

simple cases okay

so we're going to try something else and

that's what this um that's what this is

about

this is really a fancy word for using

chain rule backwards reverse chain rule

it's a version of reverse chain rule and

the way you can see that is so you've

got a four x squared here right where

the um where the that those two um

standard integrals that we've just

written on the board came from is

there's a simpler version of this

without all this on a business right

what's the very first thing that we

stated just for

a result but we'll get us this slightly

verse what would that be

one on

one

very good okay so

you got this kind of thing happening and

that's just regular old sine inverse

okay now when you have a look at that

long story i'll explain later um

when you have a look at this

okay

what this is really saying is there's

something

it's being squared and then that's

what's going to end up under here okay

but if you started without that if you

started for something like here

then of course something else will get

squared and that's where you end up okay

does that make sense so when i see that

i've got two choices the first way is i

can think about this guy and say well i

just want a single x squared there so

that's why i'll take out that factor of

two that leaves me with a single x

squared there

or alternatively i can say

i want a one there see how i've got a

one here i've got a one there and that's

exactly what i want the other thing that

needs to get muscled up with is this

okay so i say well where did that come

from here's the method i'm going to use

now so you could say this is method

number one

here comes number two okay

i will flag it right now this method the

second approach for this question is

going to be longer okay it's going to be

like why why didn't we go about this way

when we could have done this this simple

path and the answer is if your question

is not so simple

then you can't do it just in three lines

and not just it's um you can't just do

it in three lines but there's lots more

places for errors to creep in because

you start to do the numbers in your hand

it gets very confused

so here's what i'm going to suggest

instead i'm going to say let u equal 2x

so i'm going to introduce a substitution

right that's what's implied by chamber

okay often because you're getting quite

good at it now we'll just do chain rule

in our heads right but this is where it

all comes from this is what's happening

in the background

i am taking 2x because what i do is i

see this

part underneath the um in the integrand

and i say look this guy here is really

2x all squared do you see that so in

other words i'm searching for

i'm searching for this thing whatever it

is that's b squared in this case it's 2x

okay

now what i'm going to do therefore is

i'm going to substitute here

everything on this in the integrand i

want to change it into use

and therefore if everything is in terms

of u if that's the variable

then i want to instead of integrating

with respect to x

i want to integrate with respect to u

right so i'm going to change the

function i'm integrating and then i'm

going to change the variable i'm

integrating with respect to does that

make sense

so to do that

instead of dx's i want to use i need

some connection between dx and u so i'm

going to use this i can say i could

differentiate this with respect to that

right so therefore d u

on

dx this is just like dydx right except

i'll just give it a different name

in this case simple the answer is

okay

i'm gonna do one more thing here um

don't write this down

sometimes you will see i certainly see

that in um

in different places and i've said it

before myself sometimes you'll see at

this line

what's the way we would write it in this

case some people will write this

okay now just pause again like i said

don't write this down

why would people write this okay and the

answer is

this line here is what i'm going to

substitute into here right or i should

say 2x there so it'll just become 1

minus u squared okay which is exactly

what i want you see that 1 minus

something squared

and then they say well i need to get rid

of all the x's there done i need to get

rid of this dx right so a natural thing

to do is i'll take this equation here

and i'll rephrase it i'll make the

subject dx or i can flip it around you

see what i mean okay

so then what i'm going to eventually do

what that results in i'll put half to u

up there does that make

sense now

numerically everything will work

everything will come out in the watch

we've looked before at the beginning of

the inverse functions we said look d y

and d x d x and d y they really do

behave kind of like a normal fraction

right let's say cancel out and then

become one

i'm going to encourage you not to write

it like this okay and the reason why is

because let's just think back to what um

dx and y and u and all that kind of

thing where did they come from and what

do they

mean because even though they behave a

lot like a normal fraction

these on their own are normal numbers

what are they

yeah it's some infinitesimally small

thing okay now in the same way that you

know we can say something like this

okay i can say that

i can put infinity in here and i can

work with it because we understand

what's going on it's like ah denominator

is getting big whole numbers getting

small if i go all the way all the way

i'll get to zero yes so we sort of

understand what's going on here but

infinity is not a number right it's not

like i can substitute infinity in that's

the whole point of using limits and in

the same way

these guys are the opposite of

infinities we actually call them

infinitesimals right they're things that

are microscopically small so it doesn't

really make sense to say that this

microscopic small thing

which is meant to be behaving just like

zero

just like this is something that's

behaving just like infinitely large

that one is half the size of the other

it doesn't really make sense these are

not numbers that sort of can operate on

their own

this line is

not technically correct the reason we

can work with it like say here why is

this guy hanging on um seemingly by

himself is the answer is he is not by

himself what's he paired with

it's paired with this right now if you

remember the whole point of this is that

this is a sum of something times

something one of these two things meant

to come together in form

think way back to the very start of

integration

it's it's this tiny little rectangle

right do you remember that and we're

adding up an infinite series of really

really thin rectangles so the reason why

this guy seems to be there on his own is

he's the width

and he's the height right so it's like

okay these two things operate together

and here these two things operate

together because there's a ratio between

them

but these this number really shouldn't

be these numbers shouldn't be on their

own they're not meant to live like that

in the same way you can't put infinity

inside an equation you can't write this

really that's

it's not a number that can be

substituted it's an idea okay so the

same thing is happening here

i'm going to encourage you not to write

that even if you think it here i think

is a better more technical way to write

and you can you can write this number

the difference is almost nothing

i'm going to write this

okay

and the reason this is little better a

little better is because the the

d and the x they stay together it's

still a derivative

i've done this because before remember i

made dx the subject because i'm like oh

there's dx there i better substitute

here okay

here i have made one the subject the

useful thing about one is that i can put

one whoever i like right i can multiply

something by one and it doesn't change

what it is okay so therefore

i'm ready now i know right i'm ready to

do all of my substitutions at this point

i'm ready to change what the function is

that i'm integrating and i'm ready to

change what the variable is that i've

integrated with respect to

let's do it so i've got that question

over there one on

so have you known

i'll show you in some ways it doesn't

matter because multiplication is

commutative so i can rearrange however i

like because it really is this times

this so so long as i multiply by one

somewhere in the chain you'll see what

happens

okay i'm going to do one thing before i

do the substitution which is i'm going

to include this line here to make clear

why is it that i'm doing any of this

okay why did i choose 2x so this is

going to become

this okay now this line

i'm putting it directly parallel to this

one because each one is our slight

algebraic

um twist

in order to use a different standard

integral okay so in the first case the

one you suggested to me at the beginning

i'm trying to reshape it so it looks

like that does that make sense

in this case this second one i'm trying

to reshape this so it looks like this so

i've got a one minus something

underneath the square root does that

make sense okay so just re-written it

just a little bit

now having done all of this work over

here and i'd put this like on the right

hand side of my normal working i'll have

some space over there i'm going to make

all the substitutions i need to for

starters

i can see on the underneath the square

root that's just going to be 1 minus

u squared

cool that's handy

at this point i'm now going to use my

last line over there on the left hand

side

to change the variable of integration so

i'm going to do this a half d u of dx

dx okay so you see like i'm multiplying

by one so i can put this guy this is

what i mean i could put him anywhere i

like within this integral okay but

having done this because it does

function like a normal fraction

those two will cancel this is the chain

rule part right like literally do you

want the x equals that and the two

chains the two to use cancel so here

it's two dx's that cancel so now i'm

going to bring that half out the front

this is the integral of this

and you can see at this point see how

this is parallel

to what i was working out before the

half has come out the front

except the one thing that's different is

because i'm not using an x on a form x18

instead i'm using this right

yes

so the u equals 2x just hops right into

place because that's that's the fall i'm

using

okay make sense do you see how i did it

now as i'm just pointing out

this is considerably longer considerably

longer but this method is a lot more

versatile to take on harder things where

the numbers are not just going to be so

straightforward