integral of xsqrt(196-49x^2)

解答

\int x 196 - 49 x^{2} d x

- \frac{7}{3} (- x^{2} + 4)^{\frac{3}{2}} + C

求解步骤

\int x 196 - 49 x^{2} d x

使用换元积分法

= \int - \frac{u}{98} d u

提出常数:

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

= - \frac{1}{98} \cdot \int u d u

使用幂法则

= - \frac{1}{98} \cdot \frac{2}{3} u^{\frac{3}{2}}

u = 196 - 49 x^{2}

代回

= - \frac{1}{98} \cdot \frac{2}{3} (196 - 49 x^{2})^{\frac{3}{2}}

化简

- \frac{1}{98} \cdot \frac{2}{3} (196 - 49 x^{2})^{\frac{3}{2}} : - \frac{7}{3} (- x^{2} + 4)^{\frac{3}{2}}

= - \frac{7}{3} (- x^{2} + 4)^{\frac{3}{2}}

解答补常数

= - \frac{7}{3} (- x^{2} + 4)^{\frac{3}{2}} + C

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