How do you prove this integral?

Question

Prove that
∫(1/x) = ln(x)

And don't prove that the derivative of ln(x) is 1/x and then use the fundamental theorem. That's not what I'm asking for here.

Answer 1

Well, but the usual proof is just based on the fact that ln(x) is the inverse of e^x, which implies d/dx l(n)x = 1/x. Since 1/x is continuous, the conclusion follows from the fundamental theorem of Calculus.

Another possibility is just to define ln(x) = Integral (1 to x) 1/t dt, then there's nothing to prove, it's just definition.

The fact is that the evaluation of ∫(1/x) depends on the definition of the ln function. You could also define the ln function as the function such that

ln(e) = 1
ln(x y) = ln(x) + ln(y) for every x, y >0. If you suppose ln is differentiable at least one element of R+, then you conclude it's differentiable all over R+ and that d/dx ln(x) = 1/x, x >0. Then, you apply the fundamental theorem of Calculus.

Answer 2

Interesting question! I am guessing you can get the rieman sum to look like the Taylor Series, but I will have to play around with it a bit to see if I can get it.

It is a fun arithmetic exercise to show that the sum of 1/x lies between between: log(base2)n and log(base4)n. It can be done by grouping the terms by into groups of sizes increasing powers of 2.

Answer 3

Use excel.

Plot a graph of y = 1/x, and between 0 and x measure the area under the curve [Area(x)] using a large number of trapezia of arbitrarily small width w. Plot a graph of Area(x) - ln(x) as a function of w. As w tends to zero, Area(x) - ln(x) should also tend asymptotically to zero.

Answer 4

use y= 1/x or x^ -1
go to fundamental of intigreation theoram
then you can prove this



ok thanks