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受欢迎的微积分 >

integral of sin^6(x/2)

解答

\int sin^{6} (\frac{x}{2}) d x

解答

2 (- \frac{1}{6} sin^{5} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{5}{96} (- 4 sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + 3 (x - sin (x)))) + C

求解步骤

\int sin^{6} (\frac{x}{2}) d x

使用换元积分法

= \int 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 sin^{6} (u) d u

使用降次积分法

= 2 (- \frac{cos ( u ) sin ^{5} ( u )}{6} + \frac{5}{6} \cdot \int sin^{4} (u) d u)

\int sin^{4} (u) d u = - \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u))

= 2 (- \frac{cos ( u ) sin ^{5} ( u )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (u) cos (u) + \frac{3}{8} (u - \frac{1}{2} sin (2 u))))

u = \frac{x}{2}

代回

= 2 (- \frac{cos ( \frac{x}{2} ) sin ^{5} ( \frac{x}{2} )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} - \frac{1}{2} sin (2 \cdot \frac{x}{2}))))

化简

2 (- \frac{cos ( \frac{x}{2} ) sin ^{5} ( \frac{x}{2} )}{6} + \frac{5}{6} (- \frac{1}{4} sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{3}{8} (\frac{x}{2} - \frac{1}{2} sin (2 \cdot \frac{x}{2})))) : 2 (- \frac{1}{6} sin^{5} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{5}{96} (- 4 sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + 3 (x - sin (x))))

= 2 (- \frac{1}{6} sin^{5} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{5}{96} (- 4 sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + 3 (x - sin (x))))

解答补常数

= 2 (- \frac{1}{6} sin^{5} (\frac{x}{2}) cos (\frac{x}{2}) + \frac{5}{96} (- 4 sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) + 3 (x - sin (x)))) + C

作图

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