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受欢迎的微积分 >

integral from-pi to pi of sin^6(x/2)

解答

\int_{- π}^{π} sin^{6} (\frac{x}{2}) d x

解答

\frac{5 π}{8}

+1

十进制

1.96349 \dots

求解步骤

\int_{- π}^{π} sin^{6} (\frac{x}{2}) d x

使用换元积分法

= \int_{- \frac{π}{2}}^{\frac{π}{2}} sin^{6} (u) \cdot 2 d u

提出常数:

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

= 2 \cdot \int_{- \frac{π}{2}}^{\frac{π}{2}} sin^{6} (u) d u

使用降次积分法

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

\int_{- \frac{π}{2}}^{\frac{π}{2}} sin^{4} (u) d u = \frac{3 π}{8}

= 2 (- [\frac{cos ( u ) sin ^{5} ( u )}{6}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} \cdot \frac{3 π}{8})

化简

2 (- [\frac{cos ( u ) sin ^{5} ( u )}{6}]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5}{6} \cdot \frac{3 π}{8}) : 2 (- [\frac{1}{6} sin^{5} (u) cos (u)]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5 π}{16})

= 2 (- [\frac{1}{6} sin^{5} (u) cos (u)]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{5 π}{16})

带入上下限计算后得:

0

= 2 (- 0 + \frac{5 π}{16})

化简

= \frac{5 π}{8}

作图

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