I want to find the Laplace transform of the integral $$\int_0^1 (x-t)u(t)dt$$
The integral resembles a convolution but it is not, since the convolution would be $$\int_0^x (x-t)u(t)dt$$, whose Laplace transform is well known.
I tried splitting the integral as follows $$\int_0^1 (x-t)u(t)dt =\int_0^x (x-t)u(t)dt -\int_1^x (x-t)u(t)dt$$ now the first term is a convolution but the second one has a lower limit of $1$ rather than $0$, so it's not a convolution.
So what should I do to find the Laplace transform of this integral?