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Is there a way to make the first part of this look like the second picture depicted here? Also after intigration is there a way to make the highlighted posrtion not have an "ln(e)" parts and just have the exponetials and there constants?

Obviously I dont want to have to manually input this section everytime, is there some command I can use to achieve this goal?

Above is the outcome I am getting at the current time... and I used map(expand,...) to do the expansion then simplifing it goes back to the origional. 

Below is the outcome I would like to achieve...you can see that this is a different method. The problem I have is this is the right click method of expansion and I am tring to build a program that solves matrix ODE's by just inputing the initial matrix and it doing everything so i can check my answers step by step. 

So my question is, is there another method of command input to get the bottom line in the below picture?

 

 

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