integral of (cos(2x)-3)/(cos(2x))

解答

\int \frac{cos ( 2 x ) - 3}{cos ( 2 x )} d x

x - \frac{3}{2} ln ∣ tan (2 x) + sec (2 x) ∣ + C

求解步骤

\int \frac{cos ( 2 x ) - 3}{cos ( 2 x )} d x

乘开

\frac{cos ( 2 x ) - 3}{cos ( 2 x )} : 1 - \frac{3}{cos ( 2 x )}

= \int 1 - \frac{3}{cos ( 2 x )} d x

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 1 d x - \int \frac{3}{cos ( 2 x )} d x

\int 1 d x = x

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

= x - \frac{3}{2} ln ∣ tan (2 x) + sec (2 x) ∣

解答补常数

= x - \frac{3}{2} ln ∣ tan (2 x) + sec (2 x) ∣ + C

\int \frac{1}{10} e^{- \frac{x}{10}} d x

\int - tan^{2} (x) d x

\int \frac{x}{121 x ^{2} - 1} d x

d er i v a t i v e y = 2^{- x}

\int (2 w^{2} - w) e^{7 w - 1} d w