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I know how to usually do these kinds of questions but usually there is something given to substitute for x or y in F or parametric equation is given.

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    "j"? Do you mean by that $i$? – 2017-01-13
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    i and j are unit vectors i cap and j cap – 2017-01-13

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If you don't want to use Green's Theorem, then you need to concoct your own parametrization. In this case, the curve is made of 4 pieces, so the integral breaks into 4 parts. The segment from $(-1,0)$ to $(0,1)$ is given by $x=t-1$, $y=t$ for $0\leq t \leq 1$. Work out that integral and then do the other 3 segments similarly. Then add up the 4 results.

The point of such exercises is to make you appreciate Green's Theorem.

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One way: if $S$ is the given square and $\partial S$ its boundary, $$\int_{\partial S}xydx+xy^2dy\underbrace{=}_{\text{Green's Theorem}}\iint_{S}(y^2-x)dxdy=\ldots$$

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    Thankyou .... But I need to do it normally... I want to know my mistake – 2017-01-13
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    Well, in that case, write your work. – 2017-01-13
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    Iam sorry but that's too jumbled... I can try to explain you see I divided the square into 4 curves then iam solving each curve separately but iam not able to convert all the equation into x so I can take the limits of x only that's where iam stuck – 2017-01-13
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    Look at the excellent B. Goddard's answer (+1) – 2017-01-13