Expert Tips for Solving ∫e1/x/[x(x+1)2]dx | Integral Homework Help

[Sturk200]

Homework Statement

Here is the integral:

∫e^1/x/[x(x+1)²]dx

The Attempt at a Solution

I tried doing a partial fraction decomposition, but I'm not sure if that is permitted since the numerator is not a regular polynomial. Any help would be greatly appreciated! Thanks.

 

[PhotonSSBM]

Homework Statement

Here is the integral:

∫e^1/x/[x(x+1)²]dx

The Attempt at a Solution

I tried doing a partial fraction decomposition, but I'm not sure if that is permitted since the numerator is not a regular polynomial. Any help would be greatly appreciated! Thanks.

There's a tricky u-substitution you can use that's not the most obvious and it has something to do with that numerator. Can you see what it is?

 

[Sturk200]

There's a tricky u-substitution you can use that's not the most obvious and it has something to do with that numerator. Can you see what it is?

How about another hint?

 

[MidgetDwarf]

For future reference, you have to type out your try at a solution. I'll be nice this time, and hopefully I do not get a warning for helping you.

You can combine multiple techniques. You may have to preform a substitution or a algebraic manipulation. Since you tried partial fractions, yes you cannot proceed because of that e^(1/x).

Try a u-sub and tell me what you get. I actually solved this problem. It is very long. Maybe there is a short-cut but i could not see it.

 

[MidgetDwarf]

How about another hint?

Physicsnorum Physics Forums does not operate the same way google answer does. You have to actually make attempts and try. This forum is not a solutions manual.

 

[Sturk200]

For future reference, you have to type out your try at a solution. I'll be nice this time, and hopefully I do not get a warning for helping you.

You can combine multiple techniques. You may have to preform a substitution or a algebraic manipulation. Since you tried partial fractions, yes you cannot proceed because of that e^(1/x).

Try a u-sub and tell me what you get. I actually solved this problem. It is very long. Maybe there is a short-cut but i could not see it.

Thanks for your reply.

I guess I'm getting stuck pretty early in the problem. I tried letting u=1/x, then the integral turns into -e^(u)x/[(x+1)^2] du, but that is a mess. I don't really know what else to try. Maybe you can point to a step I can make to kelp me get my foot in the door of a solution?

 

[Mark44]

Thanks for your reply.

I guess I'm getting stuck pretty early in the problem. I tried letting u=1/x, then the integral turns into -e^(u)x/[(x+1)^2] du, but that is a mess. I don't really know what else to try. Maybe you can point to a step I can make to kelp me get my foot in the door of a solution?

I haven't worked it through like MidgetDwarf has, so I don't know what works. It might be that u = 1/x is a good substitution, but once you've done the substitution, your integrand should be entirely in terms of u and du -- no x terms or dx should still remain.

One thing you might try is to use partial fractions on the $\frac 1 {x(x + 1)^2}$ part. That way you could break up the integral into three integrals of the form
$$A\int \frac{e^{1/x} dx}{x} + B\int \frac{e^{1/x} dx}{x + 1} + C\int \frac{e^{1/x} dx}{(x + 1)^2}$$
I don't know if this hint is helpful. My aim is splitting up one harder integral into three that are easier.

 

[PhotonSSBM]

Thanks for your reply.

I guess I'm getting stuck pretty early in the problem. I tried letting u=1/x, then the integral turns into -e^(u)x/[(x+1)^2] du, but that is a mess. I don't really know what else to try. Maybe you can point to a step I can make to kelp me get my foot in the door of a solution?

How can you rewrite the x terms as u's using that substitution? In other words what does x equal in terms of u?

 

[Sturk200]

How can you rewrite the x terms as u's using that substitution? In other words what does x equal in terms of u?

I think I worked through the substitution as you suggested and now have: [-u*e^u]/[(u+1)^2] du. I got this using x=1/u. I think I must be missing something because this looks just as tough to me as the original one. Is there some way I should be leveraging integration by parts at this point?

 

[MidgetDwarf]

I think I worked through the substitution as you suggested and now have: [-u*e^u]/[(u+1)^2] du. I got this using x=1/u. I think I must be missing something because this looks just as tough to me as the original one. Is there some way I should be leveraging integration by parts at this point?

No, you are on the right track. What other integration techniques do you have? We have trig, partial fractions, u-sub, by parts? Which one of these will work?

and no it is not as tough as the original. You got rid of the 1/x exponent on the e.

The point is, no matter how scary the integral problems look. 90 percent of the problems in your book can be worked out without resorting to more advance methods.

It just simply comes down to noticing what techniques may work and how to algebraically manipulate the function to fit the integral formulas and techniques we already have.

 

[MidgetDwarf]

I haven't worked it through like MidgetDwarf has, so I don't know what works. It might be that u = 1/x is a good substitution, but once you've done the substitution, your integrand should be entirely in terms of u and du -- no x terms or dx should still remain.

One thing you might try is to use partial fractions on the $\frac 1 {x(x + 1)^2}$ part. That way you could break up the integral into three integrals of the form
$$A\int \frac{e^{1/x} dx}{x} + B\int \frac{e^{1/x} dx}{x + 1} + C\int \frac{e^{1/x} dx}{(x + 1)^2}$$
I don't know if this hint is helpful. My aim is splitting up one harder integral into three that are easier.

 

[Ray Vickson]

How can you rewrite the x terms as u's using that substitution? In other words what does x equal in terms of u?

If u = 1/x, isn't it really easy to get x in terms of u?

 

[Sturk200]

Alright, I have taken your hints, for which I am of course very thankful, and have now "decomposed" the u-substitution into two separate terms. That is, from [-u*e^u]/[(u+1)^2]du, I got: e^u/[(u+1)^2] - e^u/(u+1). As far as I am concerned, I have just gone from having one integral that I am incapable of solving, to having two of them .... Was it correct to do that decomposition?

 

[SammyS]

Alright, I have taken your hints, for which I am of course very thankful, and have now "decomposed" the u-substitution into two separate terms. That is, from [-u*e^u]/[(u+1)^2]du, I got: e^u/[(u+1)^2] - e^u/(u+1). As far as I am concerned, I have just gone from having one integral that I am incapable of solving, to having two of them .... Was it correct to do that decomposition?

Now, maybe let t = u+1, i.e: u = t-1 .

It looks like after some simplifying, you may need integration by parts.