integral from 0 to 3 of xsqrt(3-x)

解答

\int_{0}^{3} x 3 - x d x

- \frac{18 3}{5} + 63

十进制

4.15692 \dots

求解步骤

\int_{0}^{3} x 3 - x d x

使用换元积分法

= \int_{3}^{0} - (- u + 3) u d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{0}^{3} - (- u + 3) u d u

提出常数:

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

= - (- \int_{0}^{3} (- u + 3) u d u)

乘开

(- u + 3) u : - u^{\frac{3}{2}} + 3 u

= - (- \int_{0}^{3} - u^{\frac{3}{2}} + 3 u d u)

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} u^{\frac{3}{2}} d u + \int_{0}^{3} 3 u d u))

\int_{0}^{3} u^{\frac{3}{2}} d u = \frac{18 3}{5}

\int_{0}^{3} 3 u d u = 63

= - (- (- \frac{18 3}{5} + 63))

化简

= - \frac{18 3}{5} + 63

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