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受欢迎的微积分 >

integral of cot^3(x/2)csc^4(x/2)

解答

\int cot^{3} (\frac{x}{2}) csc^{4} (\frac{x}{2}) d x

解答

2 (- \frac{1}{6} csc^{6} (\frac{x}{2}) + \frac{1}{4} csc^{4} (\frac{x}{2})) + C

求解步骤

\int cot^{3} (\frac{x}{2}) csc^{4} (\frac{x}{2}) d x

使用换元积分法

= \int cot^{3} (u) csc^{4} (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 cot^{3} (u) csc^{4} (u) d u

使用三角恒等式改写

= 2 \cdot \int \frac{( 1 - sin ^{2} ( u ) ) cos ( u )}{sin ^{7} ( u )} d u

使用换元积分法

= 2 \cdot \int \frac{1 - v ^{2}}{v ^{7}} d v

乘开

\frac{1 - v ^{2}}{v ^{7}} : \frac{1}{v ^{7}} - \frac{1}{v ^{5}}

= 2 \cdot \int \frac{1}{v ^{7}} - \frac{1}{v ^{5}} d v

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int \frac{1}{v ^{7}} d v - \int \frac{1}{v ^{5}} d v)

\int \frac{1}{v ^{7}} d v = - \frac{1}{6 v ^{6}}

\int \frac{1}{v ^{5}} d v = - \frac{1}{4 v ^{4}}

= 2 (- \frac{1}{6 v ^{6}} - (- \frac{1}{4 v ^{4}}))

代回

= 2 (- \frac{1}{6 sin ^{6} ( \frac{x}{2} )} - (- \frac{1}{4 sin ^{4} ( \frac{x}{2} )}))

化简

2 (- \frac{1}{6 sin ^{6} ( \frac{x}{2} )} - (- \frac{1}{4 sin ^{4} ( \frac{x}{2} )})) : 2 (- \frac{1}{6} csc^{6} (\frac{x}{2}) + \frac{1}{4} csc^{4} (\frac{x}{2}))

= 2 (- \frac{1}{6} csc^{6} (\frac{x}{2}) + \frac{1}{4} csc^{4} (\frac{x}{2}))

解答补常数

= 2 (- \frac{1}{6} csc^{6} (\frac{x}{2}) + \frac{1}{4} csc^{4} (\frac{x}{2})) + C

作图

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