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受欢迎的微积分 >

integral of cot^4(xcs)c^2x

解答

\int cot^{4} (x cs) c^{2} x d x

解答

- \frac{c}{3 s} x cot^{3} (cs x) + c^{2} x^{2} + \frac{c}{s} x cot (cs x) - \frac{c ^{2} x ^{2}}{2} - \frac{4}{3 s ^{2}} ln ∣ sin (cs x) ∣ - \frac{1}{6 s ^{2}} cot^{2} (cs x) + C

求解步骤

\int cot^{4} (x cs) c^{2} x d x

提出常数:

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

= c^{2} \cdot \int cot^{4} (cs x) x d x

使用换元积分法

= c^{2} \cdot \int \frac{u cot ^{4} ( u )}{c ^{2} s ^{2}} d u

提出常数:

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

= c^{2} \frac{1}{c ^{2} s ^{2}} \cdot \int u cot^{4} (u) d u

使用分布积分法

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

\int - \frac{1}{3} cot^{3} (u) + u + cot (u) d u = \frac{u ^{2}}{2} + \frac{4}{3} ln ∣ sin (u) ∣ + \frac{1}{6} cot^{2} (u)

= c^{2} \frac{1}{c ^{2} s ^{2}} (u (- \frac{1}{3} cot^{3} (u) + u + cot (u)) - (\frac{u ^{2}}{2} + \frac{4}{3} ln ∣ sin (u) ∣ + \frac{1}{6} cot^{2} (u)))

u = cs x

代回

= c^{2} \frac{1}{c ^{2} s ^{2}} (cs x (- \frac{1}{3} cot^{3} (cs x) + cs x + cot (cs x)) - (\frac{( cs x ) ^{2}}{2} + \frac{4}{3} ln ∣ sin (cs x) ∣ + \frac{1}{6} cot^{2} (cs x)))

化简

c^{2} \frac{1}{c ^{2} s ^{2}} (cs x (- \frac{1}{3} cot^{3} (cs x) + cs x + cot (cs x)) - (\frac{( cs x ) ^{2}}{2} + \frac{4}{3} ln ∣ sin (cs x) ∣ + \frac{1}{6} cot^{2} (cs x))) : - \frac{c}{3 s} x cot^{3} (cs x) + c^{2} x^{2} + \frac{c}{s} x cot (cs x) - \frac{c ^{2} x ^{2}}{2} - \frac{4}{3 s ^{2}} ln ∣ sin (cs x) ∣ - \frac{1}{6 s ^{2}} cot^{2} (cs x)

= - \frac{c}{3 s} x cot^{3} (cs x) + c^{2} x^{2} + \frac{c}{s} x cot (cs x) - \frac{c ^{2} x ^{2}}{2} - \frac{4}{3 s ^{2}} ln ∣ sin (cs x) ∣ - \frac{1}{6 s ^{2}} cot^{2} (cs x)

解答补常数

= - \frac{c}{3 s} x cot^{3} (cs x) + c^{2} x^{2} + \frac{c}{s} x cot (cs x) - \frac{c ^{2} x ^{2}}{2} - \frac{4}{3 s ^{2}} ln ∣ sin (cs x) ∣ - \frac{1}{6 s ^{2}} cot^{2} (cs x) + C

作图

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