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受欢迎的微积分 >

integral of cot^3(3x)csc^5(3x)

解答

\int cot^{3} (3 x) csc^{5} (3 x) d x

解答

- \frac{1}{3} (\frac{1}{7} csc^{7} (3 x) - \frac{1}{5} csc^{5} (3 x)) + C

求解步骤

\int cot^{3} (3 x) csc^{5} (3 x) d x

使用换元积分法

= \int cot^{3} (u) csc^{5} (u) \frac{1}{3} d u

提出常数:

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

= \frac{1}{3} \cdot \int cot^{3} (u) csc^{5} (u) d u

使用换元积分法

= \frac{1}{3} \cdot \int - v^{4} (v^{2} - 1) d v

提出常数:

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

= \frac{1}{3} (- \int v^{4} (v^{2} - 1) d v)

乘开

v^{4} (v^{2} - 1) : v^{6} - v^{4}

= \frac{1}{3} (- \int v^{6} - v^{4} 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

= \frac{1}{3} (- (\int v^{6} d v - \int v^{4} d v))

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

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

= \frac{1}{3} (- (\frac{v ^{7}}{7} - \frac{v ^{5}}{5}))

代回

= \frac{1}{3} (- (\frac{csc ^{7} ( 3 x )}{7} - \frac{csc ^{5} ( 3 x )}{5}))

化简

\frac{1}{3} (- (\frac{csc ^{7} ( 3 x )}{7} - \frac{csc ^{5} ( 3 x )}{5})) : - \frac{1}{3} (\frac{1}{7} csc^{7} (3 x) - \frac{1}{5} csc^{5} (3 x))

= - \frac{1}{3} (\frac{1}{7} csc^{7} (3 x) - \frac{1}{5} csc^{5} (3 x))

解答补常数

= - \frac{1}{3} (\frac{1}{7} csc^{7} (3 x) - \frac{1}{5} csc^{5} (3 x)) + C

作图

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