integral of (tan(3x)-cot(3x))/(sin(3x))

解答

\int \frac{tan ( 3 x ) - cot ( 3 x )}{sin ( 3 x )} d x

\frac{1}{3} ln ∣ tan (3 x) + sec (3 x) ∣ + \frac{1}{3} csc (3 x) + C

求解步骤

\int \frac{tan ( 3 x ) - cot ( 3 x )}{sin ( 3 x )} d x

使用三角恒等式改写

= \int \frac{sin ( 3 x ) - cot ( 3 x ) cos ( 3 x )}{cos ( 3 x ) sin ( 3 x )} d x

乘开

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

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

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

\int \frac{cot ( 3 x )}{sin ( 3 x )} d x = - \frac{1}{3} csc (3 x)

= \frac{1}{3} ln ∣ tan (3 x) + sec (3 x) ∣ - (- \frac{1}{3} csc (3 x))

化简

= \frac{1}{3} ln ∣ tan (3 x) + sec (3 x) ∣ + \frac{1}{3} csc (3 x)

解答补常数

= \frac{1}{3} ln ∣ tan (3 x) + sec (3 x) ∣ + \frac{1}{3} csc (3 x) + C

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

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

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

\int \frac{x ^{2} + 2 x + 3}{x ^{2} - 6 x + 8} d x

\int 1 + u d u