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受欢迎的微积分 >

integral of (8x^3)/(sqrt(5-x^4))

解答

\int \frac{8 x ^{3}}{5 - x ^{4}} d x

解答

- 4 5 - x^{4} + C

求解步骤

\int \frac{8 x ^{3}}{5 - x ^{4}} d x

提出常数:

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

= 8 \cdot \int \frac{x ^{3}}{5 - x ^{4}} d x

使用换元积分法

= 8 \cdot \int - \frac{1}{4 u} d u

提出常数:

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

= 8 (- \frac{1}{4} \cdot \int \frac{1}{u} d u)

使用幂法则

= 8 (- \frac{1}{4} \cdot 2 u^{\frac{1}{2}})

u = 5 - x^{4}

代回

= 8 (- \frac{1}{4} \cdot 2 (5 - x^{4})^{\frac{1}{2}})

化简

8 (- \frac{1}{4} \cdot 2 (5 - x^{4})^{\frac{1}{2}}) : - 4 5 - x^{4}

= - 4 5 - x^{4}

解答补常数

= - 4 5 - x^{4} + C

作图

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