What is the integral from x/2 to 1-x/2 of (2-x-2y) ?

The integral from x/2 to 1-x/2 of (2-x-2y) is x^2-2x+1

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integral from x/2 to 1-x/2 of (2-x-2y)

\int_{\frac{x}{2}}^{1 - \frac{x}{2}} (2 - x - 2 y) d y

x^{2} - 2 x + 1

Solution steps

\int_{\frac{x}{2}}^{1 - \frac{x}{2}} 2 - x - 2 y d y

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{\frac{x}{2}}^{1 - \frac{x}{2}} 2 d y - \int_{\frac{x}{2}}^{1 - \frac{x}{2}} x d y - \int_{\frac{x}{2}}^{1 - \frac{x}{2}} 2 y d y

\int_{\frac{x}{2}}^{1 - \frac{x}{2}} 2 d y = 2 - 2 x

\int_{\frac{x}{2}}^{1 - \frac{x}{2}} x d y = x - x^{2}

\int_{\frac{x}{2}}^{1 - \frac{x}{2}} 2 y d y = 1 - x

= 2 - 2 x - (x - x^{2}) - (1 - x)

Simplify

2 - 2 x - (x - x^{2}) - (1 - x) : x^{2} - 2 x + 1

= x^{2} - 2 x + 1

What is the integral from x/2 to 1-x/2 of (2-x-2y) ?
The integral from x/2 to 1-x/2 of (2-x-2y) is x^2-2x+1