Function f(x)=1/x intergral

1. Jan 25, 2005

Aki

I have the function f(x)=1/x , and there's two asymptotes, the x-and y-axes. As x gets larger, the f(x) becomes smaller, but it is never 0. So my question is, what is the intergral from x=1 to x=infinite?

2. Jan 25, 2005

Sirus

$$\int \frac{1}{u}\,du=\ln{|u|}+C$$

3. Jan 25, 2005

digink

The derivative of ln x is 1/x so like the person stated above

$$\int (1/u)du = ln|u| + C$$

4. Jan 25, 2005

Aki

I'm sorry, but I'm so lost already. Could somebody please explain it to me? What is u?

5. Jan 25, 2005

digink

Let me put it in a simpler form, just replace u with x something you are probably more common to seeing.

Now the derivative for the ln x = 1/x.

now if you have $$\int {1/x}dx$$ you know that's the derivative of the ln of x, so you end up with that = $$ln|x| + C$$

this is just based of knowing the derivative and antiderivative of ln x, thats all you need to know.

6. Jan 25, 2005

kreil

integrals at infinity are calculated by

$$\int_1^{\infty}f(x)dx=\lim_{t{\rightarrow}\infty}\int_1^tf(x)dx$$

if you use this in combination with the info above you can calculate it

7. Jan 25, 2005

JasonRox

I never learned that yet. That's pretty cool.

8. Jan 28, 2005

Aki

so basically, there's not "number" answer to that questions? The answer is just a function?
and also where did ln(x) come from?

9. Jan 28, 2005

vincentchan

$$\int_1^{\infty} \frac{1}{x}dx$$ is undefined, or infinite.. depend on which one you feel more comfortable

where did ln x came from...hmmm... it came from [itex] \frac{d}{dx} lnx = 1/x [/tex].... so your next question is why this is true.....

assume you know product rule and the derivative of e^x is e^x itself

$$e^{\ln{x}} = x$$

$$\frac{d}{dx} e^{\ln{x}} = \frac{d}{dx} x$$

$$e^{\ln{x}} \frac{d}{dx} (\ln{x}) =1$$ --------product rule

$$x \frac{d}{dx} (\ln{x})=1$$

$$\frac{d}{dx} (\ln{x}) = \frac{1}{x}$$

so the anti-derivative of 1/x is ln(x)

10. Jan 28, 2005

Zurtex

Erm vincentchan don't you mean the chain rule?

11. Jan 28, 2005

dextercioby

Yes,it is the chain rule...Anyway,the result is correct and the method of finding it is correct as well...

Daniel.