integral of cot^7(x)

解答

\int cot^{7} (x) d x

- \frac{1}{6} cot^{6} (x) + \frac{1}{4} cot^{4} (x) - \frac{1}{2} cot^{2} (x) - ln ∣ sin (x) ∣ + C

求解步骤

\int cot^{7} (x) d x

使用降次积分法

= - \frac{cot ^{6} ( x )}{6} - \int cot^{5} (x) d x

\int cot^{5} (x) d x = - \frac{1}{4} cot^{4} (x) + \frac{1}{2} cot^{2} (x) + ln ∣ sin (x) ∣

= - \frac{cot ^{6} ( x )}{6} - (- \frac{1}{4} cot^{4} (x) + \frac{1}{2} cot^{2} (x) + ln ∣ sin (x) ∣)

化简

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

= - \frac{1}{6} cot^{6} (x) + \frac{1}{4} cot^{4} (x) - \frac{1}{2} cot^{2} (x) - ln ∣ sin (x) ∣

解答补常数

= - \frac{1}{6} cot^{6} (x) + \frac{1}{4} cot^{4} (x) - \frac{1}{2} cot^{2} (x) - ln ∣ sin (x) ∣ + C

\int 9 cos^{2} (θ) d θ

d er i v a t i v e [3 (3 x^{5} + 6)^{3}] (12 x^{3})

a re a x = 9 y^{3}, x = 23 y^{2}

\frac{\partial}{\partial z} (e^{x y})

d er i v a t i v e f (x) = (2 x + 1)^{5}