integral from 0 to 2 of-5x^3sqrt(x^2+4)

解答

\int_{0}^{2} - 5 x^{3} x^{2} + 4 d x

- \frac{64 2 + 64}{3}

十进制

- 51.50322 \dots

求解步骤

\int_{0}^{2} - 5 x^{3} x^{2} + 4 d x

提出常数:

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

= - 5 \cdot \int_{0}^{2} x^{3} x^{2} + 4 d x

使用换元积分法

= - 5 \cdot \int_{2}^{22} (u^{2} - 4) u^{2} d u

乘开

(u^{2} - 4) u^{2} : u^{4} - 4 u^{2}

= - 5 \cdot \int_{2}^{22} u^{4} - 4 u^{2} 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

= - 5 (\int_{2}^{22} u^{4} d u - \int_{2}^{22} 4 u^{2} d u)

\int_{2}^{22} u^{4} d u = \frac{128 2 - 32}{5}

\int_{2}^{22} 4 u^{2} d u = \frac{4 ( 16 2 - 8 )}{3}

= - 5 (\frac{128 2 - 32}{5} - \frac{4 ( 16 2 - 8 )}{3})

化简

- 5 (\frac{128 2 - 32}{5} - \frac{4 ( 16 2 - 8 )}{3}) : - \frac{64 2 + 64}{3}

= - \frac{64 2 + 64}{3}

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