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

解答

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

- \frac{1}{6} cot^{2} (3 x) + C

求解步骤

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

使用换元积分法

= \int - \frac{u}{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 u d u

使用幂法则

= - \frac{1}{3} \cdot \frac{u ^{2}}{2}

u = cot (3 x)

代回

= - \frac{1}{3} \cdot \frac{cot ^{2} ( 3 x )}{2}

化简

- \frac{1}{3} \cdot \frac{cot ^{2} ( 3 x )}{2} : - \frac{1}{6} cot^{2} (3 x)

= - \frac{1}{6} cot^{2} (3 x)

解答补常数

= - \frac{1}{6} cot^{2} (3 x) + C

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