Integration problem of a quotient

[rashida564]

Homework Statement

What is the integral of e^(x)2 over (x-1)²

Homework Equations

integral of (x-1)^-2 is -(x-1)^-1
and derivative of e^(x)2 is 2xe^(x)2

The Attempt at a Solution

I tried to integrate it by part but I couldn't get a solution. I want to know how to start solving this question.

 

[eys_physics]

I don't think this has an antiderivative in terms of elementary functions. However, for given limits you can compute it numerically.

 

[rashida564]

I don't think this has an antiderivative in terms of elementary functions. However, for given limits you can compute it numerically.

how someone can know if it has an anti derivative

 

[Ray Vickson]

Homework Statement

What is the integral of e^(x)2 over (x-1)²

Homework Equations

integral of (x-1)^-2 is -(x-1)^-1
and derivative of e^(x)2 is 2xe^(x)2

The Attempt at a Solution

I tried to integrate it by part but I couldn't get a solution. I want to know how to start solving this question.

I doubt that your integral can be performed in finite terms.

The simpler integral $\int e^{x^2} \, dx$ is known to be "non-elementary", which means that there is no possible finite formula for it that involves only only "ordinary functions". It does not matter if you allow a formula of 1,000,000 page length; as long as you do not use infinitely many pages you will not be able to do it. BTW: this is not a matter of nobody being smart enough to see how to do it; it is a rigorously proven theorem that it is impossible to do!

Of course, the integral exists, but the issue here is how to calculate it. There are numerous good approximations that allow us to get accurate numerical answers---even to hundreds of decimal places----so in practice we can get numbers easily enough.

 

[eys_physics]

I should had said that it doesn't have any known antiderivative. But, Mathematica could not compute it.

 

[rashida564]

I doubt that your integral can be performed in finite terms.

The simpler integral $\int e^{x^2} \, dx$ is known to be "non-elementary", which means that there is no possible finite formula for it that involves only only "ordinary functions". It does not matter if you allow a formula of 1,000,000 page length; as long as you do not use infinitely many pages you will not be able to do it. BTW: this is not a matter of nobody being smart enough to see how to do it; it is a rigorously proven theorem that it is impossible to do!

Of course, the integral exists, but the issue here is how to calculate it. There are numerous good approximations that allow us to get accurate numerical answers---even to hundreds of decimal places----so in practice we can get numbers easily enough.

So sir any integral that involves e^(x)2 can't be solved

 

[rashida564]

can I write the answer as ∑(xⁿ)/(n+1)n factorial from n=0 to infinity

 

[rashida564]

My teacher made a prank on me I've been trying to solve it for more than 6 hours

 

[Ray Vickson]

So sir any integral that involves e^(x)2 can't be solved

No, some can be solved and some cannot. The integral $\int x e^{x^2} \, dx$ is perfectly well solvable by a change of variables to $u = x^2.$

 

[Ray Vickson]

can I write the answer as ∑(xⁿ)/(n+1)n factorial from n=0 to infinity

This is not a "finite" formula, even though you can write it in a limited amount of space. When you expand it out you get a series that never ends.