Integration of: 1/(x^2) * e^(1/x)

[Ceasium]

Hi,

 

I need to integrate the following function: f(x) = 1/x² * e^1/x Where x does not equal 0. Determine a number a < 0 such that:

 

The integral of a till 0 f(x)dx = f(a)

 

What the question aks is on other words: Find a number a, such that the number a equals the surface under the graph and its position on the x-axis.

 

All that remains for me is to determine the integral of the function,

 

I have tried integration by parts, substitution with u = e^1/x , and improper intergrals, but none seemed to work. (I was working in loops, or making the integral even more difficult)

 

Can someone help me in the right direction?

 

[Cap'n Refsmmat]

Maybe try u-substituting u = 1/x.

[Ceasium]

Thanks for your reply. It got me somewhere, yet I am stuck again on the unsolvable part. I've attached my working out here, and on the second page I try to work out the integral, but then come across a problem. Can you see where my computation went wrong? Or should I try a whole different approach?

 

-edit: now attached the files

math hw 5 calc 5 (1).pdf

Math hw 5 calc 1 (2).pdf

Edited October 21, 2013 by Ceasium

[daniton]

I don't know how you do it but it is wrong.

Use u= 1/x the substitute this and dx the do the integrations it is pretty easy.

Edited October 21, 2013 by daniton

[Cap'n Refsmmat]

It is not true that if u=1/x, then

 

[math]\int \frac{e^{1/x}}{x^2} \, dx = \int u^2 e^u \, du[/math]

 

You need to figure out what du is in terms of dx.

[overtone]

Is visual recognition of a chain rule derivative an allowed suggestion? Seems like the plethora of techniques available has led to confusion - KISS.

 

You just want something that gives you that derivative - doesn't matter how you find it. Trial and error is a perfectly good approach.

[Ceasium]

I don't know how you do it but it is wrong.

Use u= 1/x the substitute this and dx the do the integrations it is pretty easy.

 

It is not true that if u=1/x, then

 

[math]\int \frac{e^{1/x}}{x^2} \, dx = \int u^2 e^u \, du[/math]

 

You need to figure out what du is in terms of dx.

 

I used your approach, and it worked, see the attached files. But if I now try to evaluate the integral on the given points, all I get is nonsense.

 

If I evaluate the integral on the given numbers, I get a non- true answer. (see attached files in my previous post).

 

Can someone see what I am doing wrong whilst evaluating the limits

[Cap'n Refsmmat]

Where did you use it? In the files attached to post #3, you did your u-substitution incorrectly, which forced you to use integration by parts. You should not have to integrate by parts.

 

Remember: if u=1/x, then du=1/x²dx. You need to account for this when you replace dx with du.

[Ceasium]

Thanks, that was the step that I needed to solve it correctly