Impossible question

[pastel yew]

How do you do this question?
find the indefinite integral of x^dx - 1

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[naive sandal]

x^dx-1=(x^dx-1)/dx *dx

Now think of the definition of the derivative

hint: consider the function f(t)=x^t

[pastel yew]

derivative is tx^t-1

[naive sandal]

Now really because here it’s with respect to t

[pastel yew]

but how does it work with an operator rather than a variable

if its with respect to t

isnt it x^t * ln(t)

[naive sandal]

Yep

For t=0

[naive sandal]

No ln(x)

[pastel yew]

no you are right

[naive sandal]

x^t * ln(x)

For t=0

[pastel yew]

so how does this work

from here

[naive sandal]

Well in this case (x^dx-1)/dx *dx = ln(x) *dx

As we have established

[pastel yew]

how is it just lnxdx

how is t = 0?

[naive sandal]

Like I said use the definition of the limit f’(t)=(f(t+dx)-f(t))/dx

( you can consider dx so small that it’s like it approaches 0)

That’s the reasoning here

[naive sandal]

use this for f(t)=x^t

And for t=0

[pastel yew]

so you get this

[naive sandal]

It’s basically the derivative of f at t=0

[pastel yew]

$\frac{x^{t + dx} - x^{t}}{dx}$

[carmine iron]

Kap

[pastel yew]

that

[naive sandal]

Yep for t=0

Here

[pastel yew]

and you are effectively taking x^t out to the front

to get that x^t * my original thing

but we are treating it as t -----> 0

so x^t = 1

but how does the ln come in now?

[naive sandal]

Yes you can look at it that way as well

Well what is the derivative of f(t)=x^t?

With respect to t

[pastel yew]

this with lnx right?

[naive sandal]

Yes

[pastel yew]

so we are basically saying that both are the derivative

[naive sandal]

Now for t=0 what is it equal to ?

[pastel yew]

so we are making them equal to each other

[naive sandal]

I guess yes basically the expression here is equal to ln(x)dx

As we’ve just established

So you need to find the antiderivative of x—>ln(x) now

[pastel yew]

thats xlnx - x

ik that

[naive sandal]

Yes

[pastel yew]

im still a bit caught up in the logic tho

so we are saying that this

$\frac{x^{dx} - 1}{dx}$

[carmine iron]

Kap

[pastel yew]

is equal to x^t

[naive sandal]

no ln(x)

It’s the derivative of t—>x^t at t=0

Which is ln(x)

[pastel yew]

let me start again

[pastel yew]

we represent the limit as dx tends to 0 as equal to the derivative of x^t with respect to t

[naive sandal]

Yes

[pastel yew]

so when we take the derivative of x^t which is x^t * lnx

when it approaches 0

the x^t ----> 1

so it becomes lnx

[naive sandal]

Yes

[pastel yew]

so we then have lnxdx

[naive sandal]

Yes

[pastel yew]

where we use IBP

with 1 and lnx

to get that the integral is equal to xlnx - x + c

[naive sandal]

Yes

Perfecto

[pastel yew]

thanks so much

i need to frame this chat

and put it on my wall

this is crazy

[naive sandal]

Lmao

[pastel yew]

thanks so much

.close

[narrow zodiac]

Unable to parse the channel name

[pastel yew]

!close

how do i close this

[naive sandal]

You’re welcome

[naive sandal]

+close

[pastel yew]

+close