Math Help - [SOLVED] transversality condition

I need some help with the transversality condition.

I have a functional to minimize over an admissible class with one free end value
here it is

$int x'(1 + t^2*x')dt$

with $x(1)= 1$

the Euler equation gives me

$(1-c)/(2t) + c_1$

the transversality condition to be satisfied at $t= 2$ is

$f_x' (2, x(2), x'(2)) = 0$

can you please show me the steps to get c and $c_1$ using the transversality and the fixed end value?

thanks for the help

The minimum is . The condition gives one equation involving and , and the transversality condition gives another.