Having trouble with a fairly simple integration

[Pandabasher]

Homework Statement

I'm in the middle of solving a D.E using integration factors, and I've ended up with an integral on the R.H.S which is as follows: \displaystyle \int \frac {e^ \frac {-1}{x}}{x^4} dx


I've tried to use integration by parts to do it, but I keep having to use it again and again. I've checked that my integrating factor is correct, and it is. I'm just having a hard time doing it. Any help is greatly appreciated. Thanks, Matt.

 

[Mark44]

Try splitting it this way:
u = x^(-2), dv = x^(-2)e^(-1/x)dx

I haven't worked it through, so can't guarantee this will work, but this is what I would try.

 

[Pandabasher]

Try splitting it this way:
u = x^(-2), dv = x^(-2)e^(-1/x)dx

I haven't worked it through, so can't guarantee this will work, but this is what I would try.

Okay, thanks. I'll give it a go.

 

[dextercioby]

If you let 1/x = t (new variable), what do you get ?

 

[Dick]

I would start by substituting u=(-1/x). The integration by parts should be straightforward. You only need to do it twice. Can you show where you are having problems?

 

[Pandabasher]

I would start by substituting u=(-1/x). The integration by parts should be straightforward. You only need to do it twice. Can you show where you are having problems?

Using \displaystyle u= \frac {-1}{x} I get \displaystyle \int e^{u} u^{2} du

Then I used integration by parts to get that equal to \displaystyle u^{2}e^{u} - \int 2u e^{u} du

Then using integration by parts again I get that equal to \displaystyle 2u e^{u} - \int 2e^{u} which is just \displaystyle 2u e^{u} - 2e^{u}

Then subbing back in and taking out a factor of \displaystyle e^u I get the answer to the first integral as \displaystyle e^{u}(u^{2}-2u+2)

And subbing u back in I get \displaystyle e^{ \frac {-1}{x}}( { \frac {1}{x^{2}}} + { \frac {2}{x}} + 2)

Does that look right?

 

[Mark44]

You can check yourself - differentiate what you ended with and you should get the integrand of your integral.

 

[Pandabasher]

You can check yourself - differentiate what you ended with and you should get the integrand of your integral.

Yeah I checked it with wolfram, its correct. Thanks for the tip, I'm not that good with integrals so I probably wouldn't have spotted that.

 

[dextercioby]

Late I am, but for the record, don't leave out the integration constant, because it shows what you're looking for: the 'family' of antiderivatives.