积分 sec^3(x)

解答

积分

sec^{3} (x)

\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣ + C

求解步骤

\int sec^{3} (x) d x

使用降次积分法

= \frac{sec ^{2} ( x ) sin ( x )}{2} + \frac{1}{2} \cdot \int sec (x) d x

\int sec (x) d x = ln ∣ tan (x) + sec (x) ∣

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

化简

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

= \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣

解答补常数

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

\int \frac{1}{( 1 + x ) ( 1 + x ^{2} )} d x

\int (\frac{2 p + 3}{( p - 1 ) ^{2}}) d p

\int \frac{arcsin ( x )}{x ( 1 - x )} d x

\int \frac{3}{3 x - 5} d x

\int 4^{5 x} d x