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I had this in my previous cats that I'm not sure whether it's really a complex analysis question, looks like a differential question with line integrals a bit

$$\int_{(1,3)}^{(4,5)} (2y+x^2)\,dx + (3x-y)\,dy$$

I was thinking the closest concept would be contour integration axis-wise i.e horizontally then vertically, such that the gradient of the line is $y=(7+2x)/3$ which when you replace in the expressions you would get something like:

$$\int_{(1,3)}^{(4,5)} (7/3+2/3x)+x^2)\,dx + \int_{(1,3)}^{(4,5)} (9/2y-21/2-y) \, dy$$

Is this the right way?

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$\newcommand{\angles}[1]{\left\langle\,{#1}\,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{\mathrm{i}} \newcommand{\iff}{\Leftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\Li}[2]{\,\mathrm{Li}_{#1}\left(\,{#2}\,\right)} \newcommand{\ol}[1]{\overline{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\ul}[1]{\underline{#1}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$

\begin{align} \color{#f00}{2} & = \partiald{\pars{2y + x^{2}}}{y} \color{#f00}{\not=} \partiald{\pars{3x - y}}{x} = \color{#f00}{3} \end{align} So, your result will be $\color{#f00}{path\ dependent}$.

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