integration problem

Can anyone solve this integration? I need the answer with respect to t.
Attachment 26579

Hi !
This integral cannot be expressed as a combination of a finite number of usual functions. The analytical solving involves special functions, such as Ei(x) and will lead to a wery complicated formula.
Of course it is possible to obtain this formula, but it would take a lot of time. Probably, this time would be lost without benefit, because it would be too difficult to use the big formula for further work.
That is the reason why I suggest to treat your problem with numerical calculus methods or with approximative analytical methods (limited series expansions depending on the physical range)

Thank you. I was wondering if it has some simpler way. But it seems that I will face with complex numbers any way.
I am trying to model numerical solution of this problem in matlab and it takes a long time for each step to give an answer.
Can you give me a hint about limited series expansion? I do not have any idea about it. I have just tried the numerical way.

Using series approximations is also a numerical way. For series expansion, see :
Taylor Series -- from Wolfram MathWorld
Approximation of a function in a limited range can be done thanks to a few terms of the series instead of the complete series. Of course, the deviations are all the larger as the number of terms chosen is small and as the range of variation of the function is large. Often it is necessary to cut in several domains with different series for each one. All depends on the wanted accuracy. Spending a lot of time in preliminary studies is generaly necessary to be sure that the chosen limited series are convenient.
Moreover, it is not sure that this method requires less computing time that direct numerical integration if an efficient integrator package is used. Again, it depends on the accuracy needed.

If you have a fast computation package for numerical computation of Laplace transforms, another way is to use the formula shown in attachment
:

Thanks a lot. It was a great idea. I will try it and show you the answer if you are willing. Thanks again