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受欢迎的微积分 >

integral of (3x)/(\sqrt[3]{10-x^2)}

解答

\int \frac{3 x}{3 10 - x ^{2}} d x

解答

- \frac{9}{4} (10 - x^{2})^{\frac{2}{3}} + C

求解步骤

\int \frac{3 x}{3 10 - x ^{2}} 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}{3 10 - x ^{2}} d x

使用换元积分法

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

提出常数:

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

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

使用幂法则

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

u = 10 - x^{2}

代回

= 3 (- \frac{1}{2} \cdot \frac{3}{2} (10 - x^{2})^{\frac{2}{3}})

化简

3 (- \frac{1}{2} \cdot \frac{3}{2} (10 - x^{2})^{\frac{2}{3}}) : - \frac{9}{4} (10 - x^{2})^{\frac{2}{3}}

= - \frac{9}{4} (10 - x^{2})^{\frac{2}{3}}

解答补常数

= - \frac{9}{4} (10 - x^{2})^{\frac{2}{3}} + C

作图

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