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受欢迎的微积分 >

integral of sec(x)tan^5(x)

解答

\int sec (x) tan^{5} (x) d x

解答

sec (x) - \frac{2 sec ^{3} ( x )}{3} + \frac{sec ^{5} ( x )}{5} + C

求解步骤

\int sec (x) tan^{5} (x) d x

使用三角恒等式改写

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

使用换元积分法

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

乘开

(- 1 + u^{2})^{2} : 1 - 2 u^{2} + u^{4}

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

\int 1 d u = u

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

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

= u - \frac{2 u ^{3}}{3} + \frac{u ^{5}}{5}

u = sec (x)

代回

= sec (x) - \frac{2 sec ^{3} ( x )}{3} + \frac{sec ^{5} ( x )}{5}

解答补常数

= sec (x) - \frac{2 sec ^{3} ( x )}{3} + \frac{sec ^{5} ( x )}{5} + C

作图

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