integral of ((tan(x))/(cot(x)))^3

解答

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

\frac{tan ^{5} ( x )}{5} - \frac{tan ^{3} ( x )}{3} + tan (x) - arctan (tan (x)) + C

求解步骤

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

使用三角恒等式改写

= \int tan^{6} (x) d x

使用换元积分法

= \int \frac{u ^{6}}{1 + u ^{2}} d u

长除

\frac{u ^{6}}{1 + u ^{2}} : u^{4} - u^{2} + 1 - \frac{1}{u ^{2} + 1}

= \int u^{4} - u^{2} + 1 - \frac{1}{u ^{2} + 1} d u

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

= \int u^{4} d u - \int u^{2} d u + \int 1 d u - \int \frac{1}{u ^{2} + 1} d u

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

\int u^{2} d u = \frac{u ^{3}}{3}

\int 1 d u = u

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

= \frac{u ^{5}}{5} - \frac{u ^{3}}{3} + u - arctan (u)

u = tan (x)

代回

= \frac{tan ^{5} ( x )}{5} - \frac{tan ^{3} ( x )}{3} + tan (x) - arctan (tan (x))

解答补常数

= \frac{tan ^{5} ( x )}{5} - \frac{tan ^{3} ( x )}{3} + tan (x) - arctan (tan (x)) + C

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

x \to 3 lim (x^{2} - 5 x + 6)

(y^{3} - 2 x^{2} y) d x + (2 x^{3}) d y = 0

x \to - 6 lim (\frac{( 4 x + 3 ) ^{\frac{2}{3}} - 9}{x ^{2} - 36})

\int \frac{π}{3}