I am looking at the implicit euler integration scheme for an equation $dx/dt = xt + 1$
How do you arrive at the second line from the first?


I don't think the second line does follow from the first, for with
$x_{k + 1} = x_k + h(x_{k + 1} t_{k + 1} + 1), \tag 1$
we find
$x_{k + 1} = x_k + hx_{k + 1} t_{k + 1} + h, \tag 2$
whence
$x_{k + 1} - hx_{k + 1} t_{k + 1} = x_k + h, \tag 3$
or 
$x_{k + 1}(1 - ht_{k + 1}) = x_k + h; \tag 4$
thus, provided $1 - ht_{k + 1} \ne 0$,
$x_{k + 1} = (1 - ht_{k + 1})^{-1}(x_k + h) \ne x_k + h(1 - ht_{k + 1})^{-1}. \tag 5$
