so good
okay i'm going to show you guys two substitutions to integral of tan(x)*sec^2(x) let's do the
obvious one first for this one right here let me just say that u equal to tangent x
how come because i know right away when i differentiate this i get the derivative of tangent
x which is secant squared x dx right and you see i can take this integral into the u world right away
this is the integral tangent x is my u right this is my u right here and secant square x times dx
that's my du so we are just integrating u in the u world right away of course this is u to the
first power you add one to the power and divided by the new power so we get one half u squared
u is tangent x so just put that down right here so finally we have one half tangent square x and we
are done so put on plus c like this this is the first result that we can have for that integral
and now let's check this out remember this is the integral of tangent x times secant square x
so technically we have two functions namely tangent x and also secant x right
is it possible for us to let u equal to secant x well in this case yes but
we have to just do it carefully because you have to know the derivative secant x is what
tangent x times secant x right so now this is how i would do it right here
i'm going to first look at this integral as a following i will look at this as the integral
let me break the secant square x the part as following let me write it down a secant x first
and then we still have this tangent x right let me just put that down
and then times another secant x this secant x times this secant x we get the secant squared x
and then let me put down the dx right here all right and now i'm going to let u
equal to this secant x because now you see du will be secant x tangent x dx isn't it and you will see
this is equal to the integral the secant x is the u right it's this u right here that's the secant x
and tangent x secant x dx this right here is my d u so i'm just integrating u
in the u world as well do this right here we get one half u squared right in this case u
is secant x so our result here is one half secant square x we are done we can also put down plus c
like this okay now you may be wondering this is the answer that we got earlier involving tangent
this is the result that we got right now involving secant which one it's actually correct well maybe
they're both correct well maybe we did something wrong
well in this case they're actually both correct but they look different because of the constant c
so what we should be doing is that this right here should be c1 and this right here should be c2
the result of the integration could be off by a constant right once again the result of the
integration could be off by a constant so if you do it this way so you get a constant c1 maybe
your friend did it this way well they have another constant you have to label them as c1 and c2 and
now here is the explanation why they are actually correct so this is the idea of off by a constant
let me just put this is equal to that first for you guys so this is kind of like my check
on the left hand side let me put down one half tangent square x plus z1 and then on
the right hand side let me put down one half secant square x plus c2 and you will see
let me just go ahead multiply everything by two because we don't like fractions okay
let's multiply everything by two hopefully you guys don't mind this at all and you will have
one half times two which is just one so that's tangent squared x two times c one right plus two c
one plus i means this is equal to this times that which is just one we have c can square x
two times c two right so plus c2 like this and now this is what i would like to do
let me subtract 2c2 on both sides right here 2 times c2 right here sorry anyway let's do it
again let me minus 2 c 2 on both sides so i'll put down minus 2 c 2 2 c 1 right here right
okay so this right here is out on the left hand side i have tangent square x on the right hand
side we have secant squared x and this right here is just a constant it's just a number
this right here let me write it down o in blue why not plus 2 c 2 minus 2 c 1 all right remember
c 1 is a number c 2 is also number and just to be fancier we can say that constants right so
c1 is a constant if you have minus c2 this is still a constant this is a constant times two
is a constant constant minus another constant all together is the constant and you may be wondering
what in the world is that constant do you guys recognize what this voice should be well we know
right this is the famous identity that you guys should know as well we know that tangent square x
this has to be equal to secant square x and then what -1 this is an identity well now
this match this match so what is this this and that has to be the same isn't it
so in other words 2 c 2 minus 2 c 1 is nothing but just negative 1 right so this is the idea we may
get different looking results but they're just off by a constant if you differentiate this you
will get that for the answer for the original if you differentiate that you also get that for the
derivative so they are both the anti-derivative of the integrand anyway that's it right