Question: Integrate the function f(z) = x$^2$ + i xy from A(1 , 1) to B(2 , 4) along the curve x = t , y = t$^2$

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$ f(z) = x^2 + ixy \\ $; A(1,1,) to B(2,4)

x = t; y = t$^2$; dx = dt; dy = 2t dt

dz = dx + idy

= dt + i(2t dt)

= (1+2it) dt

$ f(z) = t^2 + i(t)(t^2) \\ = t^2 + it^3 $

$ \int_A^B f(z) \,\, dx = \int_1^2 (t^2 + it^3)(dt + i(2t) dt) \\ = \int_1^2 (t^2 + it^3)(1+2it) \,\, dt \\ = \int_0^1 (t^2 + 2it^3+it^3-2t^4) \,\, dt \\ = \int_0^1 (t^2 + 3it^3-2t^4) \,\, dt \\ = [\frac{t^3}{3} + \frac{3it^4}{4} - \frac{2t^5}{5}]_1^2 \\ = \frac{7}{3} + \frac{48i}{4} - \frac{3i}{4} - \frac{64}{5} + \frac{2}{5} \\ = \frac{7}{3} - \frac{62}{5} + \frac{45i}{4} \\ = \frac{-151}{15} + \frac{45i}{4} $

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