Question relating to solving equations with Green's functions

An easy question hopefully. I have the solution to a differential equation

$y'+2ty=g(t)$

in the form

$y=\int^t_{t_{0}}g(\tau)G(t,\tau)d\tau$

where $G(t,\tau)$ is greens function thus

$y=\int^t_{t_{0}}g(\tau)e^{\tau^2-t^2}d\tau$

if I want to solve it for

$g(t) &=t, \ 1 \leq t \leq 2$

would I get

$y=\int^2_{1} \tau e^{\tau^2-4}d\tau$

I am just not sure about replacing the t in the equation with the upper limit.

Thanks guys/girls