integral from 0 to a of 7xsqrt(a^2-x^2)

解答

\int_{0}^{a} 7 x a^{2} - x^{2} d x

\frac{7 ( a ^{2} ) ^{\frac{3}{2}}}{3}

求解步骤

\int_{0}^{a} 7 x a^{2} - x^{2} d x

提出常数:

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

= 7 \cdot \int_{0}^{a} x a^{2} - x^{2} d x

使用换元积分法

= 7 \cdot \int_{a^{2}}^{0} - \frac{u}{2} d u

提出常数:

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

= 7 (- \frac{1}{2} \cdot \int_{a^{2}}^{0} u d u)

使用幂法则

= 7 (- \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{a^{2}}^{0})

化简

7 (- \frac{1}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{a^{2}}^{0}) : - \frac{7}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{a^{2}}^{0}

= - \frac{7}{2} [\frac{2}{3} u^{\frac{3}{2}}]_{a^{2}}^{0}

带入上下限计算后得:

- \frac{2}{3} (a^{2})^{\frac{3}{2}}

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

化简

= \frac{7 ( a ^{2} ) ^{\frac{3}{2}}}{3}

roo t s \frac{a x}{x ^{2} + a ^{2}}

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