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受欢迎的微积分 >

integral of (xsqrt(7-x^2))

解答

\int (x 7 - x^{2}) d x

解答

- \frac{1}{3} (7 - x^{2})^{\frac{3}{2}} + C

求解步骤

\int x 7 - x^{2} d x

使用换元积分法

= \int - \frac{u}{2} d u

提出常数:

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

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

使用幂法则

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

u = 7 - x^{2}

代回

= - \frac{1}{2} \cdot \frac{2}{3} (7 - x^{2})^{\frac{3}{2}}

化简

- \frac{1}{2} \cdot \frac{2}{3} (7 - x^{2})^{\frac{3}{2}} : - \frac{1}{3} (7 - x^{2})^{\frac{3}{2}}

= - \frac{1}{3} (7 - x^{2})^{\frac{3}{2}}

解答补常数

= - \frac{1}{3} (7 - x^{2})^{\frac{3}{2}} + C

作图

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