integral from 0 to pi/3 of sec^5(x)

解答

\int_{0}^{\frac{π}{3}} sec^{5} (x) d x

23 + \frac{3}{4} (3 + \frac{1}{2} ln (2 + 3))

十进制

5.25699 \dots

求解步骤

\int_{0}^{\frac{π}{3}} sec^{5} (x) d x

使用降次积分法

= [\frac{sec ^{4} ( x ) sin ( x )}{4}]_{0}^{\frac{π}{3}} + \frac{3}{4} \cdot \int_{0}^{\frac{π}{3}} sec^{3} (x) d x

\int_{0}^{\frac{π}{3}} sec^{3} (x) d x = 3 + \frac{1}{2} ln (2 + 3)

= [\frac{sec ^{4} ( x ) sin ( x )}{4}]_{0}^{\frac{π}{3}} + \frac{3}{4} (3 + \frac{1}{2} ln (2 + 3))

带入上下限计算后得:

23

= 23 + \frac{3}{4} (3 + \frac{1}{2} ln (2 + 3))

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