integral from 2-y to y of 2y(2-y)

解答

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

2 (4 (y - 1) - \frac{2 y ^{3} - 6 y ^{2} + 12 y - 8}{3})

求解步骤

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

提出常数:

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

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

乘开

y (2 - y) : 2 y - y^{2}

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

使用积分加法定则:

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

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

\int_{2 - y}^{y} 2 y d y = 4 (y - 1)

\int_{2 - y}^{y} y^{2} d y = \frac{2 y ^{3} - 6 y ^{2} + 12 y - 8}{3}

= 2 (4 (y - 1) - \frac{2 y ^{3} - 6 y ^{2} + 12 y - 8}{3})

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