integral of cot^3(3x)

解答

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

\frac{1}{3} (- \frac{cot ^{2} ( 3 x )}{2} - ln ∣ sin (3 x) ∣) + C

求解步骤

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

使用换元积分法

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

使用降次积分法

= \frac{1}{3} (- \frac{cot ^{2} ( u )}{2} - \int cot (u) d u)

\int cot (u) d u = ln ∣ sin (u) ∣

= \frac{1}{3} (- \frac{cot ^{2} ( u )}{2} - ln ∣ sin (u) ∣)

u = 3 x

代回

= \frac{1}{3} (- \frac{cot ^{2} ( 3 x )}{2} - ln ∣ sin (3 x) ∣)

解答补常数

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

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