Show that the line integral is independent of path by finding a function f such that ? f = f. c 2xe? ydx (2y ? x2e? ydy, c is any path from (1, 0 to (4, 1

I'm reading this as



with .

The value of the integral will be independent of the path if we can find a function that satisfies the gradient equation above.

You have



Integrate with respect to . You get




Differentiate with respect to . You get





Integrate both sides with respect to to arrive at





So you have



The gradient is continuous for all , so the fundamental theorem of calculus applies, and so the value of the integral, regardless of the path taken, is