integral from 0 to pi/2 of sin^8(x)

解答

\int_{0}^{\frac{π}{2}} sin^{8} (x) d x

\frac{35 π}{256}

十进制

0.42951 \dots

求解步骤

\int_{0}^{\frac{π}{2}} sin^{8} (x) d x

使用降次积分法

= - [\frac{cos ( x ) sin ^{7} ( x )}{8}]_{0}^{\frac{π}{2}} + \frac{7}{8} \cdot \int_{0}^{\frac{π}{2}} sin^{6} (x) d x

\int_{0}^{\frac{π}{2}} sin^{6} (x) d x = \frac{5 π}{32}

= - [\frac{cos ( x ) sin ^{7} ( x )}{8}]_{0}^{\frac{π}{2}} + \frac{7}{8} \cdot \frac{5 π}{32}

化简

- [\frac{cos ( x ) sin ^{7} ( x )}{8}]_{0}^{\frac{π}{2}} + \frac{7}{8} \cdot \frac{5 π}{32} : - [\frac{1}{8} sin^{7} (x) cos (x)]_{0}^{\frac{π}{2}} + \frac{35 π}{256}

= - [\frac{1}{8} sin^{7} (x) cos (x)]_{0}^{\frac{π}{2}} + \frac{35 π}{256}

带入上下限计算后得:

0

= - 0 + \frac{35 π}{256}

化简

= \frac{35 π}{256}

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