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受欢迎的微积分 >

integral of (tan^3(2x))(sec^3(2x))(1)

解答

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

解答

\frac{sec ^{5} ( 2 x )}{10} - \frac{sec ^{3} ( 2 x )}{6} + C

求解步骤

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

化简

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

使用三角恒等式改写

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

使用换元积分法

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

乘开

\frac{u ^{2} ( u ^{2} - 1 )}{2} : \frac{u ^{4}}{2} - \frac{u ^{2}}{2}

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

\int \frac{u ^{4}}{2} d u = \frac{u ^{5}}{10}

\int \frac{u ^{2}}{2} d u = \frac{u ^{3}}{6}

= \frac{u ^{5}}{10} - \frac{u ^{3}}{6}

u = sec (2 x)

代回

= \frac{sec ^{5} ( 2 x )}{10} - \frac{sec ^{3} ( 2 x )}{6}

解答补常数

= \frac{sec ^{5} ( 2 x )}{10} - \frac{sec ^{3} ( 2 x )}{6} + C

作图

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