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Differentiate w.r.t. y
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\int x-y\mathrm{d}x
Evaluate the indefinite integral first.
\int x\mathrm{d}x+\int -y\mathrm{d}x
Integrate the sum term by term.
\int x\mathrm{d}x-\int y\mathrm{d}x
Factor out the constant in each of the terms.
\frac{x^{2}}{2}-\int y\mathrm{d}x
Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}.
\frac{x^{2}}{2}-yx
Find the integral of y using the table of common integrals rule \int a\mathrm{d}x=ax.
\frac{1^{2}}{2}-y-\left(\frac{0^{2}}{2}-y\times 0\right)
The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.
\frac{1}{2}-y
Simplify.