integral of 1/(cos^5(x))

解答

\int \frac{1}{cos ^{5} ( x )} d x

\frac{sec ^{4} ( x ) sin ( x )}{4} + \frac{3}{4} (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣) + C

求解步骤

\int \frac{1}{cos ^{5} ( x )} d x

使用三角恒等式改写

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

使用降次积分法

= \frac{sec ^{4} ( x ) sin ( x )}{4} + \frac{3}{4} \cdot \int sec^{3} (x) d x

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

= \frac{sec ^{4} ( x ) sin ( x )}{4} + \frac{3}{4} (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣)

解答补常数

= \frac{sec ^{4} ( x ) sin ( x )}{4} + \frac{3}{4} (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣) + C

d er i v a t i v e \frac{2}{x ^{\frac{1}{3}}} - 2

2 y^{^{'}} + t y = 7

\frac{d}{d x} (\frac{3 x - 1}{2 x + 5})

d er i v a t i v e f (x) = 7^{x}

l a pl a ce t r an s f or m \frac{2}{3} t^{3} \cdot e^{- 8 t}