What is the integral from 0 to 1-x of x-x^2-xy ?

The integral from 0 to 1-x of x-x^2-xy is x^3-2x^2+x-(x(-x+1)^2)/2

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integral from 0 to 1-x of x-x^2-xy

\int_{0}^{1 - x} x - x^{2} - x y d y

x^{3} - 2 x^{2} + x - \frac{x ( - x + 1 ) ^{2}}{2}

Solution steps

\int_{0}^{1 - x} x - x^{2} - x 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_{0}^{1 - x} x d y - \int_{0}^{1 - x} x^{2} d y - \int_{0}^{1 - x} x y d y

\int_{0}^{1 - x} x d y = x (- x + 1)

\int_{0}^{1 - x} x^{2} d y = x^{2} (- x + 1)

\int_{0}^{1 - x} x y d y = \frac{x ( - x + 1 ) ^{2}}{2}

= x (- x + 1) - x^{2} (- x + 1) - \frac{x ( - x + 1 ) ^{2}}{2}

Simplify

x (- x + 1) - x^{2} (- x + 1) - \frac{x ( - x + 1 ) ^{2}}{2} : x^{3} - 2 x^{2} + x - \frac{x ( - x + 1 ) ^{2}}{2}

= x^{3} - 2 x^{2} + x - \frac{x ( - x + 1 ) ^{2}}{2}

\int_{2}^{5} (x) d x

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\int_{0}^{4} \frac{6}{7 x + 5} d x

\int_{0}^{5} x ln (x) d x

What is the integral from 0 to 1-x of x-x^2-xy ?
The integral from 0 to 1-x of x-x^2-xy is x^3-2x^2+x-(x(-x+1)^2)/2