What is the integral from y^2 to 2-y of (6x+2y^2) ?

The integral from y^2 to 2-y of (6x+2y^2) is -5y^4-2y^3+7y^2-12y+12

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integral from y^2 to 2-y of (6x+2y^2)

\int_{y^{2}}^{2 - y} (6 x + 2 y^{2}) d x

- 5 y^{4} - 2 y^{3} + 7 y^{2} - 12 y + 12

Solution steps

\int_{y^{2}}^{2 - y} 6 x + 2 y^{2} d x

Apply the Sum Rule:

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

= \int_{y^{2}}^{2 - y} 2 y^{2} d x + \int_{y^{2}}^{2 - y} 6 x d x

\int_{y^{2}}^{2 - y} 2 y^{2} d x = 2 y^{2} (2 - y) - 2 y^{4}

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

= 2 y^{2} (2 - y) - 2 y^{4} + 3 ((- y + 2)^{2} - y^{4})

Simplify

2 y^{2} (2 - y) - 2 y^{4} + 3 ((- y + 2)^{2} - y^{4}) : - 5 y^{4} - 2 y^{3} + 7 y^{2} - 12 y + 12

= - 5 y^{4} - 2 y^{3} + 7 y^{2} - 12 y + 12

What is the integral from y^2 to 2-y of (6x+2y^2) ?
The integral from y^2 to 2-y of (6x+2y^2) is -5y^4-2y^3+7y^2-12y+12