Integral of 1/x

A quick afternoon integral, ep4.

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bprp #shorts

[theimmux]

I feel like the significance of the natural log and e isn't all that clear to a lot of people.

[riksmith]

Dont care about the math...just look at his skill to write with two marker in one hand!!!

[elreturner]

My favorite thing about this combo of derivative and antiderivative is that you can use it to explain analytic continuation without explaining analytic continuation

[taranmellacheruvu]

You can do it in the reverse direction by applying the inverse derivative theorem. To prove that:

Let f^-1(x) = g(x)
x = f(f^-1(x)) = f(g(x))
1 = f’(g(x))*g’(x)
g’(x) = 1/f’(g(x))

Now, for the real thing:

Let g(x)=lnx. Then, f(x)=e^x and f’(x)=e^x.

g’(x) = 1/e^(lnx)
g’(x) = 1/x

Therefore, the derivative of ln(x) is 1/x.

[tomasito_]

I love this proof. My question however is why x = e^t and not any other value.

[tracyrooks]

Short but clear explanation! Well done! 🤗
Creative Math with Ching-Hui

[elaceaceak]

Love love those shorts the original blackpenredpen I wasn't able to watch all the videos were 1 h

[jangy]

I was gonna say to add the absolute value but at the end you mentioned it lol

[hendrilibrata]

I have no Idea at all but still watched till the end

[jabunapg]

Isn't that the same as implicit differentiation of lnx

[Ricardo_S]

I was looking for this explanation, thanks

[blizzard]

How do you know that you have to substitute e^t though

[TheNetkrot]

beautiful thank you ... your the best.

[maharshijoshi]

You could do it by Applying limit n -> -1 (integral(x^n))

[krityanshutiwari]

Its because diffrentiation of lnx is 1/x

[dr.rahulgupta]

Thanks for putting x=e^t as I have suggested earlier. DrRahul Rohtak Haryana India

[charitsfachrurizalkusumara]

Can someone answer me why is "dx = e^t dt", not "dx = de^t" ?

[lbmiewy]

This equation is very, very, very easy

[JoachimFavre]

It's funny, at my calculs lessons we defined ln(x) to be the definite integral from 1 to x of 1/x, and then exp(x) to be its inverse. After this, we defined general exponentiation (what a^b means when b belongs to the real numbers ; a^b = exp[b*ln(a)]) and then saw that exp(x) = e^x.
I am curious, how do you define general exponentiatiation if ln(x) is not defined to be an antiderivative of 1/x ? :p

[Arjun-fyjy]

Nice proof. I have a question though. In this method, we have set x = e^t, meaning x will never be a negative quantity. Then why is a |x| necessary?