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

解答

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

\frac{1}{16 ( 5 - x ^{4} ) ^{4}} + C

求解步骤

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

使用换元积分法

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

提出常数:

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

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

使用幂法则

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

u = 5 - x^{4}

代回

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

化简

- \frac{1}{4} (- \frac{1}{4 ( 5 - x ^{4} ) ^{4}}) : \frac{1}{16 ( 5 - x ^{4} ) ^{4}}

= \frac{1}{16 ( 5 - x ^{4} ) ^{4}}

解答补常数

= \frac{1}{16 ( 5 - x ^{4} ) ^{4}} + C

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