$$x'=\frac{t-x}t, t>0$$

I tried to solve it by:

$x'=1-\frac{x}t$

$\int dx=\int(1-\frac{x}t)dt$

$x=t-x\ln t+C$

$x=\frac{t}{1+lnt} +C$

But this doesn't seem right. What am I doing wrong? Also I'm supposed to determine the maximal interval around t=1, but how do I get rid of $C$?