math, line integral

For path A, the equation is r(t)=ti+tj (0<=t<=1)

r'(t)=i+j

∫F(r(t)).r'(t)dt [from 0 to 1]
=∫(ti+tj)(i+j)dt
=∫2t dt
=1

For path B, the equation is r(t)=ti+t^2j (0<=t<=1)

r'(t)=i+2tj

∫F(r(t)).r'(t)dt [from 0 to 1]
=∫(ti+t^2j)(i+2tj)dt
=∫t+2t^3 dt
=1

For path C, the equation is r1(t)=ti (0<=t<=1) r2(t)=tj

r1'(t)=i r2'(t)=j

∫F(r(t)).r1'(t)+F(r(t)).r2'(t)dt [from 0 to 1]
=∫(ti)(i)+(tj)(j) dt
=1

shows that the integration is path-independent