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受欢迎的微积分 >

integral of-5tan^5(4x)sec^6(4x)

解答

\int - 5 tan^{5} (4 x) sec^{6} (4 x) d x

解答

- 5 (\frac{sec ^{10} ( 4 x )}{40} - \frac{sec ^{8} ( 4 x )}{16} + \frac{sec ^{6} ( 4 x )}{24}) + C

求解步骤

\int - 5 tan^{5} (4 x) sec^{6} (4 x) d x

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - 5 \cdot \int tan^{5} (4 x) sec^{6} (4 x) d x

使用三角恒等式改写

= - 5 \cdot \int (- 1 + sec^{2} (4 x))^{2} tan (4 x) sec^{6} (4 x) d x

使用换元积分法

= - 5 \cdot \int \frac{u ^{5} ( u ^{2} - 1 ) ^{2}}{4} d u

乘开

\frac{u ^{5} ( u ^{2} - 1 ) ^{2}}{4} : \frac{u ^{9}}{4} - \frac{u ^{7}}{2} + \frac{u ^{5}}{4}

= - 5 \cdot \int \frac{u ^{9}}{4} - \frac{u ^{7}}{2} + \frac{u ^{5}}{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

= - 5 (\int \frac{u ^{9}}{4} d u - \int \frac{u ^{7}}{2} d u + \int \frac{u ^{5}}{4} d u)

\int \frac{u ^{9}}{4} d u = \frac{u ^{10}}{40}

\int \frac{u ^{7}}{2} d u = \frac{u ^{8}}{16}

\int \frac{u ^{5}}{4} d u = \frac{u ^{6}}{24}

= - 5 (\frac{u ^{10}}{40} - \frac{u ^{8}}{16} + \frac{u ^{6}}{24})

u = sec (4 x)

代回

= - 5 (\frac{sec ^{10} ( 4 x )}{40} - \frac{sec ^{8} ( 4 x )}{16} + \frac{sec ^{6} ( 4 x )}{24})

解答补常数

= - 5 (\frac{sec ^{10} ( 4 x )}{40} - \frac{sec ^{8} ( 4 x )}{16} + \frac{sec ^{6} ( 4 x )}{24}) + C

作图

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