Solve Difficult Integral: ∫ex t-2 dt

[Prof. 27]

Homework Statement

Hi, I'm doing a variation of parameters problem for my differential equations class. It requires solving the integral:

∫e^x t^-2 dt

I am sure my professor did not give me an impossible integral and that there is some algebraic "trick" to solving it, but despite going through several iterations of integration by parts I am unable to find it (I have encountered similar problems before but my memory of them is fuzzy).

Homework Equations

None

The Attempt at a Solution

Several Integration by parts attempts. I looked for a cancellation.

 

[LCKurtz]

Homework Statement

Hi, I'm doing a variation of parameters problem for my differential equations class. It requires solving the integral:

∫e^x t^-2 dt

Both $x$ and $t$ in there?

I am sure my professor did not give me an impossible integral and that there is some algebraic "trick" to solving it, but despite going through several iterations of integration by parts I am unable to find it (I have encountered similar problems before but my memory of them is fuzzy).

Homework Equations

None

The Attempt at a Solution

Several Integration by parts attempts. I looked for a cancellation.

Please give us a statement of the original problem and your work so far. How do we know your integral is correct?

 

[Mark44]

Homework Statement

Hi, I'm doing a variation of parameters problem for my differential equations class. It requires solving the integral:

∫e^x t^-2 dt

I am sure my professor did not give me an impossible integral and that there is some algebraic "trick" to solving it, but despite going through several iterations of integration by parts I am unable to find it (I have encountered similar problems before but my memory of them is fuzzy).

Homework Equations

None

The Attempt at a Solution

Several Integration by parts attempts. I looked for a cancellation.

If you have written the integral correctly, it's a very simple one to evaluate. Here e^x can be treated as a constant.

 

[Prof. 27]

Oh I'm so sorry! I mis-wrote the integral. It is:

∫e^-x2 x^-2 dt

 

[Ray Vickson]

Oh I'm so sorry! I mis-wrote the integral. It is:

∫e^-x2 x^-2 dt

If you mean $\int e^{-x^2} x^{-2} \, dt$, that is easy: it is $e^{-x^2} x^{-2} \int dt = e^{-x^2}x^{-2} (t+C)$. If you mean $\int e^{-x^2}x^{-2} \, dx$, that is a different matter entirely. The integral is do-able in terms of the so-called error function.

On the other hand, if in the first form above the $x$ is a function of $t$, the integral may be intractable for certain functions $x = x(t)$.