Is there a closed form for this?

[EngWiPy]

Hi,

I have this integral:

$$\int_0^∞ \ln(1+x)\,e^{-x}\,dx$$

Is there any closed form expression for it?

Thanks

 

[Samy_A]

Hi,

I have this integral:

$$\int_0^∞ \ln(1+x)\,e^{-x}\,dx$$

Is there any closed form expression for it?

Thanks

Integration by parts shows that the result contains the Exponential integral function Ei evaluated in -1.
Depends of whether you call that a closed form expression.

 

[EngWiPy]

Integration by parts shows that the result contains the Exponential integral function Ei evaluated in -1.
Depends of whether you call that a closed form expression.

No, I need it in basic functions that have simple derivatives.

 

[Samy_A]

No, I need it in basic functions that have simple derivatives.

Well, the derivative of Ei is $\frac{e^x}{x}$.

 

[EngWiPy]

Integration by parts shows that the result contains the Exponential integral function Ei evaluated in -1.
Depends of whether you call that a closed form expression.

OK, thanks. Does $e^{-x}$ goes to zero faster than $\ln(1+x)$ goes to infility as $x→∞$?

 

[Samy_A]

OK, thanks. Does $e^{-x}$ goes to zero faster than $\ln(1+x)$ goes to infility as $x→∞$?

Yes. If not your integral wouldn't give a finite result. $e^{-x}$ also goes to zero faster than any polynomial goes to infinity as $x→+∞$.

 

[EngWiPy]

OK. I worked the analysis, and it won't work with me. I have something like this

$$\int_0^∞ \ln(1+s\,x)\,e^{-x}\,dx$$

and I need to find

$$\frac{\partial}{\partial s}\int_0^∞ \ln(1+s\,x)\,e^{-x}\,dx$$

from which I need to find $s$. The integral is a complicated function of the argument $s$.

 

[S.G. Janssens]

OK. I worked the analysis, and it won't work with me. I have something like this

$$\int_0^∞ \ln(1+s\,x)\,e^{-x}\,dx$$

and I need to find

$$\frac{\partial}{\partial s}\int_0^∞ \ln(1+s\,x)\,e^{-x}\,dx$$

from which I need to find $s$. The integral is a complicated function of the argument $s$.

See https://www.physicsforums.com/threads/derivative-of-integral.853211/

As discussed there, you actually need to find the Fréchet derivative of the functional acting on the function s. I have made some comments on this in that topic, but I don't think you found them very helpful, so I will rest my case. However, I wanted to point out that this problem was already discussed.

 

[EngWiPy]

See https://www.physicsforums.com/threads/derivative-of-integral.853211/

As discussed there, you actually need to find the Fréchet derivative of the functional acting on the function s. I have made some comments on this in that topic, but I don't think you found them very helpful, so I will rest my case. However, I wanted to point out that this problem was already discussed.

It is not that I found your comments unhelpful, but I just wanted to know enough to do the analysis, without going into the details. If I follow the same logic as the paper I attached there, and applied it to my problem here, I will have the p.d.f of $x$ appearing in the derivative. How to handle this when I need some numerical results?