Calculate the curve integral

Calculate the curve integral respect to arc length (integral from C ds/(x^2+y^2+1)^1/2
where C- line AB A(0,1) B(2,0)

I assume you mean the straight line between those points.
The line is y = 1 - x/2 where 0 < x < 2.
dy/dx = - 1/2
The arc length formula is
s = integral sqrt[1 + (dy/dx)^2] dx
so ds = sqrt[1 + (dy/dx)^2] dx
= sqrt[1 + (- 1/2)^2] dx
= sqrt (5/4) dx
The curve integral w.r.t. arc length is
integral ds/sqrt(x^2 + y^2 + 1)
C

= integral (x = 0 ---> 2) sqrt (5/4) dx / sqrt [x^2 + (1 - x/2)^2 + 1]
= integral (x = 0 ---> 2) sqrt (5/4) dx / sqrt [(5/4)x^2 - x + 2]
= integral (x = 0 ---> 2) dx / sqrt [x^2 - (4/5)x + 8/5]
= integral (x = 0 ---> 2) dx / sqrt [(x - 2/5)^2 + (6/5)^2]

And the rest is just ordinary calculus which you can probably do yourself.