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受欢迎的微积分 >

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

解答

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

解答

\frac{2}{sec ( x )} + \frac{2}{3} sec^{\frac{3}{2}} (x) + C

求解步骤

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

使用三角恒等式改写

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

使用换元积分法

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

乘开

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

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

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

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

= - (- \frac{2}{u}) + \frac{2}{3} u^{\frac{3}{2}}

u = sec (x)

代回

= - (- \frac{2}{sec ( x )}) + \frac{2}{3} sec^{\frac{3}{2}} (x)

化简

= \frac{2}{sec ( x )} + \frac{2}{3} sec^{\frac{3}{2}} (x)

解答补常数

= \frac{2}{sec ( x )} + \frac{2}{3} sec^{\frac{3}{2}} (x) + C

作图

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