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

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Hi,

I need to integrate the following function: f(x) = 1/x2 * e1/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 = e1/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?

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Maybe try u-substituting u = 1/x.

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

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

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

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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.

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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

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

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

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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.

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

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