What is the integral from x to 3x-x^2 of x^2-xy ?

The integral from x to 3x-x^2 of x^2-xy is -x^4+2x^3-(x(x^4-6x^3+8x^2))/2

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integral from x to 3x-x^2 of x^2-xy

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

- x^{4} + 2 x^{3} - \frac{x ( x ^{4} - 6 x ^{3} + 8 x ^{2} )}{2}

Solution steps

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

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

\int_{x}^{- x^{2} + 3 x} x y d y = \frac{x ( x ^{4} - 6 x ^{3} + 8 x ^{2} )}{2}

= - x^{4} + 2 x^{3} - \frac{x ( x ^{4} - 6 x ^{3} + 8 x ^{2} )}{2}

\int_{0}^{π} π sin (n x) d x

\int_{2}^{3} \frac{3 x - 2}{x ^{3} - x ^{2}} d x

\int_{- 2}^{2} x sin (\frac{nπ x}{2}) d x

\int_{4}^{10} \frac{1}{x - 2} d x

\int_{5}^{2} \frac{( 12 - x ^{3} )}{x ^{4}} d x

What is the integral from x to 3x-x^2 of x^2-xy ?
The integral from x to 3x-x^2 of x^2-xy is -x^4+2x^3-(x(x^4-6x^3+8x^2))/2