Answer
$\int A(t)dt=\begin{bmatrix}
e^t & -e^{-t}\\
2e^t & -5e^{-t} \\
\end{bmatrix}$
Work Step by Step
Given: $A(t)=\begin{bmatrix}
e^t & e^{-t}\\
2e^t & 5e^{-t} \\
\end{bmatrix}$
The antiderivative of the matrix function is given by:
$\int^b_aA(t)dt=\int^b_aa_{ij}(t)dt$
Hence here, $\int A(t)dt=\int\begin{bmatrix}
e^t & e^{-t}\\
2e^t & 5e^{-t} \\
\end{bmatrix}dt=\begin{bmatrix}
e^t & -e^{-t}\\
2e^t & -5e^{-t} \\
\end{bmatrix}$
