How to solve an integral with an exponential and function.

Question

How do I solve the integral $ \int \limits _0 ^t e^x y(x) \Bbb d x$? I know the integral of $y(x)$ would result in $\frac {Y(s)} s$.

Answer 1

We can write that integral as

$$\int ^\infty _0 y(x)\big(u(x)-u(x-t)\big)e^{-sx}dx$$

Where $u(x)$ is the unit step function and $s=-1$. The laplace transform of a product, is the convolution of the laplace transforms.

$$Y(s)*\frac{1-e^{-st}}{s}\bigg|_{s=-1}=\int^\infty_{-\infty}Y(-1-s)\frac{1-e^{-st}}{s}ds$$

Nothing more can be said with the given information