integral of tan^3(x)sec^{1/2}(x)

解答

\int tan^{3} (x) sec^{\frac{1}{2}} (x) d x

- 2 sec (x) + \frac{2}{5} sec^{\frac{5}{2}} (x) + C

求解步骤

\int tan^{3} (x) sec (x) d x

使用三角恒等式改写

= \int (- 1 + sec^{2} (x)) tan (x) sec (x) d x

使用换元积分法

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

乘开

\frac{- 1 + u ^{2}}{u} : - \frac{1}{u} + u^{\frac{3}{2}}

= \int - \frac{1}{u} + u^{\frac{3}{2}} 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 \frac{1}{u} d u + \int u^{\frac{3}{2}} d u

\int \frac{1}{u} d u = 2 u

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

= - 2 u + \frac{2}{5} u^{\frac{5}{2}}

u = sec (x)

代回

= - 2 sec (x) + \frac{2}{5} sec^{\frac{5}{2}} (x)

解答补常数

= - 2 sec (x) + \frac{2}{5} sec^{\frac{5}{2}} (x) + C

x \to 0 lim (\frac{1}{e ^{x}})

x \to 6 lim (\frac{x}{x ^{2} - 36})

\frac{d x}{d t} + x = - e^{t}

\frac{d}{d x} ((3 t^{2} - t + a^{2})^{101})

x \to + 3 + lim (\frac{7}{x - 3})