Given: $$\int_0^t e^{-x}Be^{(A+B)x}dx$$ where $A,B$ are matrices in $\mathbb{R}^{n \times n}$
How do you compute the definite integral?
I attempted to apply integration by parts, however I kept going in circles do to $e^x$
That is, I let $$\text{d}v = e^{-x}dx$$ and $$u = Be^{(A+B)x}$$
Is there a simple trick I should be applying to avoid this?
Thanks!
Extension:
What if: $$\int_0^t e^{-Ax}Be^{(A+B)x}dx$$ where $A,B$ are general matrices in $\mathbb{R}^{n \times n}$
Here's a simple trick: $$ \int_0^t e^{-x}Be^{(A+B)x}dx = \int_0^t B(e^{-x}\cdot I)e^{(A+B)x}dx = \int_0^t Be^{-Ix}e^{(A+B)x}dx = \int_0^t Be^{(A+B-I)x}dx. $$