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受欢迎的微积分 >

integral of tan^5(x)sec^{10}(x)

解答

\int tan^{5} (x) sec^{10} (x) d x

解答

\frac{tan ^{6} ( x )}{6} + \frac{tan ^{8} ( x )}{2} + \frac{3 tan ^{10} ( x )}{5} + \frac{tan ^{12} ( x )}{3} + \frac{tan ^{14} ( x )}{14} + C

求解步骤

\int tan^{5} (x) sec^{10} (x) d x

使用三角恒等式改写

= \int sec^{2} (x) (1 + tan^{2} (x))^{4} tan^{5} (x) d x

使用换元积分法

= \int u^{5} (1 + u^{2})^{4} d u

乘开

u^{5} (1 + u^{2})^{4} : u^{5} + 4 u^{7} + 6 u^{9} + 4 u^{11} + u^{13}

= \int u^{5} + 4 u^{7} + 6 u^{9} + 4 u^{11} + u^{13} 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^{5} d u + \int 4 u^{7} d u + \int 6 u^{9} d u + \int 4 u^{11} d u + \int u^{13} d u

\int u^{5} d u = \frac{u ^{6}}{6}

\int 4 u^{7} d u = \frac{u ^{8}}{2}

\int 6 u^{9} d u = \frac{3 u ^{10}}{5}

\int 4 u^{11} d u = \frac{u ^{12}}{3}

\int u^{13} d u = \frac{u ^{14}}{14}

= \frac{u ^{6}}{6} + \frac{u ^{8}}{2} + \frac{3 u ^{10}}{5} + \frac{u ^{12}}{3} + \frac{u ^{14}}{14}

u = tan (x)

代回

= \frac{tan ^{6} ( x )}{6} + \frac{tan ^{8} ( x )}{2} + \frac{3 tan ^{10} ( x )}{5} + \frac{tan ^{12} ( x )}{3} + \frac{tan ^{14} ( x )}{14}

解答补常数

= \frac{tan ^{6} ( x )}{6} + \frac{tan ^{8} ( x )}{2} + \frac{3 tan ^{10} ( x )}{5} + \frac{tan ^{12} ( x )}{3} + \frac{tan ^{14} ( x )}{14} + C

作图

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