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Problem:

Before I get into the problem, my professor is trying to have us define both the logarithmic and exponential functions, so we are not allowed to use things we know about logs or the exponential function, which is why I'm having trouble.

For $ x \in (0,\infty)$, define $L(x)=\int_{1}^x 1/t dt $

a) Why does the integral exist? By Theorem 30.1 (a theorem in our textbook), since L(x) is monotone, L(x) is integrable.

b) Why is L(x) differentiable and what is its derivative?

This is the one I'm most having trouble with. How exactly do I show that this function is differentiable? I know that since it is integrable, then it is uniformly continuous and that the Fundamental Theorem of Calculus gives me that $L'(x)=l(x)$, but I don't know how to state this in a way that is "proof"

c) Show that $L(1/x)=-L(x)$, in other words $\int_{1}^{1\over x}=-\int_{1}^{x}$

I feel like something that shows the following and then rearranging the terms would be best, but I'm pretty lost. $$\int_{1}^{1\over x} -\int_{1}^x = \int_{x}^x; define \int_{x}^x=0$$

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3 Answers 3

For (b), just referencing the Fundamental Theorem of Calculus is a perfectly cromulent proof. You don't have to say any special magic words in order to imbue it with proofiness, as long as you point out that each of the assumptions of you textbook's version of FTC are satisfied.

For (c), your definition of integrals should allow you to prove $$ \int_{pa}^{pb} f(t)\,dt = \frac 1p \int_a^b f(pt)\,dt \quad\text{and}\quad \int_a^b qf(t)\,dt = q\int_a^b g(t)\,dt $$ Judicious choices of $p$ and $q$ will then allow you to show $\int_{1/x}^1 \frac{dt}{t}=\int_1^x \frac{dt}{t}$.

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You just embiggened my vocabulary. –  David Mitra Dec 12 '12 at 1:29
    
Do you have any hints for what q and p should be? I've been banging my head on this for a while now! –  Jonathon Dec 12 '12 at 1:43
    
@Ray: Hint 1: They can depend on $x$. Hint 2: Choose $p$ such that when you multiply the limits by it, $\int_{1/x}^1$ becomes $\int_1^x$. –  Henning Makholm Dec 12 '12 at 14:47

For (a), it seems to me that you need to say something about the function $f(t)=1/t$ on intervals of the form $[1,x]$ (for $x>1$) and $[x,1]$ (for $x<1$) in order to deduce something about $L(x)$ - ie that it exists. There's a flaw in your logic if you are quoting a property of $L(x)$ itself in order to prove that it exists.

Part (b) is indeed an application of the fundamental theorem of calculus. Using the notation of this summary of the theorem, try to identify $f$ anf $F$ in your case, and check that the required properties of $f$ are satisfied.

For part (c), in the integral $\displaystyle{\int_1^{1/x}\frac{dt}{t}}$, try the substitution $\bar{t}=1/t$.

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For differentiability, you're correct, all that is required is saying that $\frac{1}{x}$ is continuous on $(0,\infty)$so by FTC $L'(x)$ exists and is $\frac{1}{x}$.

For the last part, instead of doing that, try a change of variables in $\int_1^x\frac{1}{t}dt$ by letting $u = \frac{1}{t}$ and see what you get.

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