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

The integral from-x to 2x-x^2 of (x+y) is -x^3+3x^2+(x^4-4x^3+3x^2)/2

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integral from-x to 2x-x^2 of (x+y)

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

- x^{3} + 3 x^{2} + \frac{x ^{4} - 4 x ^{3} + 3 x ^{2}}{2}

Solution steps

\int_{- x}^{2 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_{- x}^{- x^{2} + 2 x} x d y + \int_{- x}^{- x^{2} + 2 x} y d y

\int_{- x}^{- x^{2} + 2 x} x d y = - x^{3} + 3 x^{2}

\int_{- x}^{- x^{2} + 2 x} y d y = \frac{x ^{4} - 4 x ^{3} + 3 x ^{2}}{2}

= - x^{3} + 3 x^{2} + \frac{x ^{4} - 4 x ^{3} + 3 x ^{2}}{2}

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What is the integral from-x to 2x-x^2 of (x+y) ?
The integral from-x to 2x-x^2 of (x+y) is -x^3+3x^2+(x^4-4x^3+3x^2)/2