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受欢迎的微积分 >

integral of (3x^2)/((8-3x^3)^5)

解答

\int \frac{3 x ^{2}}{( 8 - 3 x ^{3} ) ^{5}} d x

解答

\frac{1}{12 ( - 3 x ^{3} + 8 ) ^{4}} + C

求解步骤

\int \frac{3 x ^{2}}{( 8 - 3 x ^{3} ) ^{5}} d x

提出常数:

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

= 3 \cdot \int \frac{x ^{2}}{( 8 - 3 x ^{3} ) ^{5}} d x

使用换元积分法

= 3 \cdot \int - \frac{1}{9 u ^{5}} d u

提出常数:

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

= 3 (- \frac{1}{9} \cdot \int \frac{1}{u ^{5}} d u)

使用幂法则

= 3 (- \frac{1}{9} (- \frac{1}{4 u ^{4}}))

u = 8 - 3 x^{3}

代回

= 3 (- \frac{1}{9} (- \frac{1}{4 ( 8 - 3 x ^{3} ) ^{4}}))

化简

3 (- \frac{1}{9} (- \frac{1}{4 ( 8 - 3 x ^{3} ) ^{4}})) : \frac{1}{12 ( - 3 x ^{3} + 8 ) ^{4}}

= \frac{1}{12 ( - 3 x ^{3} + 8 ) ^{4}}

解答补常数

= \frac{1}{12 ( - 3 x ^{3} + 8 ) ^{4}} + C

作图

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