using Feynamn's technique twice for the integral of sin^3(x)/x^3 from 0 to inf
[Music]
starting putting T here next to the X so we get integral going from 0 to Infinity
TX over x to the third power like so then DX here we go take the derivative with respect
to T so we will first get I prime of t and for this right here I will just do the derivative
of you guys we go from 0 to Infinity differentiate this we get three and then bring that power
to the foreign and then PSI square and then TX and then multiplied by the derivative of
sine TX which will give us cosine TX and then multiply by the derivative of the inside TX
in the t word row so we have to multiply by X chain rule from that part and then over
x to the third power and nicely enough this is not cancel what x squared ah almost yeah
[Music]
okay sine of x squared I don't really know what to do next hmm but you see the theory
is that if we differentiate this with respect to T we can squeeze our X and then this x
can cancel with an x on the pattern so if you can somehow do this again it would be
really nice yes you will but you really don't want to differentiate this right now because
it's a product of crazy stuff right so maybe let's do some identity business before we
um differentiate this again so okay I'm well use this uh um let's see I'll use the identity
three I'll put this as 1 minus
all right let's let's deal with cosine right now cosine square of TX yeah and then cosine
of TX good and then over I Wrong call it
over x square DX [Music] the street all the way at the front no problem zero to Infinity
cosine TX
that's good and then minus cosine to the third power TX
shall we really be differentiating this now [Music]
if I differentiate this we get three cosine square of t x and then
times negative sine of TX and times x and oh sorry x squared and then that can cancel
with one of the X on the bottom that's good and then
the problem here is that here you see we will maybe radius 1 minus sign square and we'll
end up with like one minus sign to the third power that's I don't see it's a good trade-off
so let's try to do what we did before that is the power reduction formula not for integral
but just the regular Central safe business so focus on that and reduce the power and
see and hope for the best [Music]
so how do we go about this let's say we have just that part cosine
of TX times cosine square of TX okay this right here is oh there are two ways to go
apart and remember one is you can write a cosine triple angle a formula and then it
has the third power part and isolate that
so I want to do that or do I want to do this [Music] um [Music] I I will do this don't
worry all right so this right here is equal to this time so I'll give you guys some more
flavor for the formula approaches and all that anyway this right here is cosine of TX
what should we do with that 1 minus sine squared no that does not really help let's use the
poor reduction formula for this namely one half times one plus cosine of two input is
t x
parentheses parenthesis yes and then we can see this Times line is one half cosine of
TX so that's that's good that's very good and then one half and we have cosine times
that right so it's cosine of TX times cosine of 2tx and now the product of these two things
is equal to one half cosine of first minus second so TX minus 2tx is negative DX times
no Plus
yeah plus um the second part is cosine of this thing plus nothing which is 3tx yeah
[Music] so [Music]
oh let me let me combine this right here first so we have that one half right here so this
is one over four so this is one over four yeah so one over four and then plus that is
three over four and then we have cosine of TX because this right here is the same as
cosine negative T thanks to Cosi being an even function and then this thing right here
is one over four times tassels plus one over four cosine of three t x [Music]
so I'm going to do this right here this is three integral 0 to Infinity cosine of TX
minus all that stuff which is three over four and then we have cosine of TX
Plus 1 over 4 cosine of 3tx
and all over x square and keep in mind we're just using trigger identities for this all
right now
this and that can be combined it one minus three over four so this is one over four and
then we have cosine of TX yeah and then it's a minus be really careful with that but the
point is the one over four one over can be fatter the all the way to the front so we
have three over four and an integral and then we have zero to infinity and then let me just
have cosine TX minus
a wonderful in the front already cosine of three t x over X the eggs all right now based
on this let's just do this in your head I double Prime of t differentiate this differentiate
that we still have the three over four at the front we still have the integral going
from 0 to Infinity take the derivative of this we get negative sine of TX and then multiply
by X yeah and then take the derivative of that is hmm originally was minus but the reality
of cosines [Music] negative so it's plus and then we have differential negative cosine
we get past design 3tx yes and then the derivative of the inside with respect to T we have to
multiply by 3x yeah and now let's split this so first we have negative three over four
integral going from 0 to infinity and then we have sign of TX over just X because this
and that canceled and we can close this nicely next two over four times three it's nine over
four so plus 9 over 4 and then we integrate from 0 to infinity and then we have sine of
TX over X and DX yes all right for this part you let T do a u sub let U equal to t x and
then right here we should just say some somewhere let's say t is greater than or equal to zero
okay
uh it should be okay there's no violation that we just focus on that if T is less than
zero thanks for thanks to sign being R function just put a negative in the front anyway that's
what we have and um yeah you can work this out you will see you it's going to give you
in fact you can have any constant multiple this is just going to give you pi over 2.
negative three over four times this thing is pi over 2. likewise plus 9 over 4 and that
will give you pi over 2 as well all this is 3 pi over four [Music] this is not the answer
yet
all right check this out
I double Prime of T is this crazy integral yeah and then you see that it's always going
to be 3 pi over 4. now we have to go back to I of T so I Prime of T right we have to
integrate this right here first which is going to be the integral of what of this result
which is our 3 pi over 4. in the T world
yeah like that and of course this right here will give us 3 pi over 4 T but don't forget
the plus c now how do we figure out a c we'll have to look back to ielt heaping this guy
so what do we do again pick an easy number for T so let T equals zero then we can see
that I'm not going to show you the order right I'll just kind of follow along here and here
son of Zero thanks to that so integrating Zero from zero to Infinity is zero equal put
the zero here is zero plus C so C is equal to zero yeah hey now we have this guy I want
to get to I of T which is the integral of this guy which is 3 pi over 4 T in the T world
and when we integrate that [Music] uh [Music] we get that's that's finished here we get
3 pi over eight t-square yes yes that one and then yeah and then Plus C let's just say
this is C1 c one c one and lastly this is C2 okay so how do we figure out what C2 is
again let's look at L of T which is this right here and we have this thing and I'm just going
to let t equal to zero this thing in blue is zero
equal distinct when T is equal to 0 is 0 plus C2 so C2 equal to zero ladies and gentlemen
I of T which is this guy integral going from zero to Infinity of PSI to the third power
t x over x to the third power DX is equal to 3 pi over eight t squared yes I'm not kidding
[Music] and when T is equal to one that's exactly why we want it so now let's t equal
to one we get the integral going from 0 to Infinity PSI to the third power of just X
over x to the third power DX is equal to these things one right so we have just three pi
over eight [Music]