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受欢迎的微积分 >

integral of 5x*\sqrt[3]{1-x^2}

解答

\int 5 x \cdot 3 1 - x^{2} d x

解答

- \frac{15}{8} (1 - x^{2})^{\frac{4}{3}} + C

求解步骤

\int 5 x 3 1 - x^{2} d x

提出常数:

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

= 5 \cdot \int x 3 1 - x^{2} d x

使用换元积分法

= 5 \cdot \int - \frac{3 u}{2} d u

提出常数:

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

= 5 (- \frac{1}{2} \cdot \int 3 u d u)

使用幂法则

= 5 (- \frac{1}{2} \cdot \frac{3}{4} u^{\frac{4}{3}})

u = 1 - x^{2}

代回

= 5 (- \frac{1}{2} \cdot \frac{3}{4} (1 - x^{2})^{\frac{4}{3}})

化简

5 (- \frac{1}{2} \cdot \frac{3}{4} (1 - x^{2})^{\frac{4}{3}}) : - \frac{15}{8} (1 - x^{2})^{\frac{4}{3}}

= - \frac{15}{8} (1 - x^{2})^{\frac{4}{3}}

解答补常数

= - \frac{15}{8} (1 - x^{2})^{\frac{4}{3}} + C

作图

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