integral from 0 to pi/2 of cos^6(x)

解答

\int_{0}^{\frac{π}{2}} cos^{6} (x) d x

\frac{5 π}{32}

十进制

0.49087 \dots

求解步骤

\int_{0}^{\frac{π}{2}} cos^{6} (x) d x

使用降次积分法

= [\frac{sin ( x ) cos ^{5} ( x )}{6}]_{0}^{\frac{π}{2}} + \frac{5}{6} \cdot \int_{0}^{\frac{π}{2}} cos^{4} (x) d x

\int_{0}^{\frac{π}{2}} cos^{4} (x) d x = \frac{3 π}{16}

= [\frac{sin ( x ) cos ^{5} ( x )}{6}]_{0}^{\frac{π}{2}} + \frac{5}{6} \cdot \frac{3 π}{16}

化简

[\frac{sin ( x ) cos ^{5} ( x )}{6}]_{0}^{\frac{π}{2}} + \frac{5}{6} \cdot \frac{3 π}{16} : [\frac{1}{6} cos^{5} (x) sin (x)]_{0}^{\frac{π}{2}} + \frac{5 π}{32}

= [\frac{1}{6} cos^{5} (x) sin (x)]_{0}^{\frac{π}{2}} + \frac{5 π}{32}

带入上下限计算后得:

0

= 0 + \frac{5 π}{32}

化简

= \frac{5 π}{32}

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