integral of x^4(9-x^5)^{1/4}

解答

\int x^{4} (9 - x^{5})^{\frac{1}{4}} d x

- \frac{4}{25} (9 - x^{5})^{\frac{5}{4}} + C

求解步骤

\int x^{4} (9 - x^{5})^{\frac{1}{4}} d x

使用换元积分法

= \int - \frac{u ^{\frac{1}{4}}}{5} d u

提出常数:

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

= - \frac{1}{5} \cdot \int u^{\frac{1}{4}} d u

使用幂法则

= - \frac{1}{5} \cdot \frac{4}{5} u^{\frac{5}{4}}

u = 9 - x^{5}

代回

= - \frac{1}{5} \cdot \frac{4}{5} (9 - x^{5})^{\frac{5}{4}}

化简

- \frac{1}{5} \cdot \frac{4}{5} (9 - x^{5})^{\frac{5}{4}} : - \frac{4}{25} (9 - x^{5})^{\frac{5}{4}}

= - \frac{4}{25} (9 - x^{5})^{\frac{5}{4}}

解答补常数

= - \frac{4}{25} (9 - x^{5})^{\frac{5}{4}} + C

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