Elementary ODEs matrix integration help

[EngageEngage]

Homework Statement

I'm trying to understand the Variation of Parameters in ODEs and I came up to this following expression which i cannot solve:


$${2\,{e}^{-t}{e}^{-3\,t}\choose {e}^{-t}{e}^{-3\,t}} \int {\,{e}^{t} {e}^{\,t}\choose {e}^{3t}{2e}^{-3\,t}} {10\,\cos \left( t \right) \choose 2\,{e}^{-t}}$$

Can I just integrate each individual component or must I use matrix multiplication first? If anyone could help me I would appreciate it greatly. I'm not sure how to even start on this so I don't have any work to show.

 

[HallsofIvy]

It's not at all clear what those mean. Do you mean
$$\left(\begin{array}{cc}2e^{-t} & e^{-3t} \\ e^{-t} & e^{-3t}\end{array}\right)\int \left(\begin{array}{cc}e^t & e^t \\ e^{3t} & 2e^{-3t}\end{array}\right)\left(\begin{array}{cc}10 & cos(t) \\ 2 & e^{-t}\end{array}\right) dt$$
(Click on the LaTex to see the code I used.)

To answer your question, yes, you must multiply before integrating: $\int f(x)g(x) dx$ is NOT $(\int f(x)dx)(\int g(x)dx)$.