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受欢迎的微积分 >

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

解答

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

解答

\frac{1}{2 ( 2 - x ^{3} ) ^{2}} + C

求解步骤

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

使用换元积分法

= 3 \cdot \int \frac{1}{3 ( 2 - u ) ^{3}} d u

提出常数:

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

= 3 \cdot \frac{1}{3} \cdot \int \frac{1}{( 2 - u ) ^{3}} d u

使用换元积分法

= 3 \cdot \frac{1}{3} \cdot \int - \frac{1}{v ^{3}} d v

提出常数:

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

= 3 \cdot \frac{1}{3} (- \int \frac{1}{v ^{3}} d v)

使用幂法则

= 3 \cdot \frac{1}{3} (- (- \frac{1}{2 v ^{2}}))

代回

= 3 \cdot \frac{1}{3} (- (- \frac{1}{2 ( 2 - x ^{3} ) ^{2}}))

化简

3 \cdot \frac{1}{3} (- (- \frac{1}{2 ( 2 - x ^{3} ) ^{2}})) : \frac{1}{2 ( 2 - x ^{3} ) ^{2}}

= \frac{1}{2 ( 2 - x ^{3} ) ^{2}}

解答补常数

= \frac{1}{2 ( 2 - x ^{3} ) ^{2}} + C

作图

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