integral from 0 to 3 of ((30-y)-y^{1/3})

解答

\int_{0}^{3} ((30 - y) - y^{\frac{1}{3}}) d y

\frac{171}{2} - \frac{9 \cdot 3 ^{\frac{1}{3}}}{4}

十进制

82.25493 \dots

求解步骤

\int_{0}^{3} 30 - y - y^{\frac{1}{3}} 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

= \int_{0}^{3} 30 d y - \int_{0}^{3} y d y - \int_{0}^{3} y^{\frac{1}{3}} d y

\int_{0}^{3} 30 d y = 90

\int_{0}^{3} y d y = \frac{9}{2}

\int_{0}^{3} y^{\frac{1}{3}} d y = \frac{9 \cdot 3 ^{\frac{1}{3}}}{4}

= 90 - \frac{9}{2} - \frac{9 \cdot 3 ^{\frac{1}{3}}}{4}

合并分式

90 - \frac{9}{2} : \frac{171}{2}

= \frac{171}{2} - \frac{9 \cdot 3 ^{\frac{1}{3}}}{4}

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