What is the integral from 1-y to-y^2+y+1 of 2y ?

The integral from 1-y to-y^2+y+1 of 2y is y^4-2y^3-2y^2+4y

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integral from 1-y to-y^2+y+1 of 2y

\int_{1 - y}^{- y^{2} + y + 1} 2 y d y

y^{4} - 2 y^{3} - 2 y^{2} + 4 y

Solution steps

\int_{1 - y}^{- y^{2} + y + 1} 2 y d y

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int_{1 - y}^{- y^{2} + y + 1} y d y

Apply the Power Rule

: [\frac{y ^{2}}{2}]_{1 - y}^{- y^{2} + y + 1}

= 2 [\frac{y ^{2}}{2}]_{1 - y}^{- y^{2} + y + 1}

Compute the boundaries:

\frac{( - y ^{2} + y + 1 ) ^{2}}{2} - \frac{( 1 - y ) ^{2}}{2}

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

Simplify

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

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What is the integral from 1-y to-y^2+y+1 of 2y ?
The integral from 1-y to-y^2+y+1 of 2y is y^4-2y^3-2y^2+4y