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受欢迎的微积分 >

integral of (9x)/((3-x^2)^3)

解答

\int \frac{9 x}{( 3 - x ^{2} ) ^{3}} d x

解答

\frac{9}{4 ( 3 - x ^{2} ) ^{2}} + C

求解步骤

\int \frac{9 x}{( 3 - x ^{2} ) ^{3}} d x

提出常数:

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

= 9 \cdot \int \frac{x}{( 3 - x ^{2} ) ^{3}} d x

使用换元积分法

= 9 \cdot \int - \frac{1}{2 u ^{3}} d u

提出常数:

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

= 9 (- \frac{1}{2} \cdot \int \frac{1}{u ^{3}} d u)

使用幂法则

= 9 (- \frac{1}{2} (- \frac{1}{2 u ^{2}}))

u = 3 - x^{2}

代回

= 9 (- \frac{1}{2} (- \frac{1}{2 ( 3 - x ^{2} ) ^{2}}))

化简

9 (- \frac{1}{2} (- \frac{1}{2 ( 3 - x ^{2} ) ^{2}})) : \frac{9}{4 ( 3 - x ^{2} ) ^{2}}

= \frac{9}{4 ( 3 - x ^{2} ) ^{2}}

解答补常数

= \frac{9}{4 ( 3 - x ^{2} ) ^{2}} + C

作图

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