integral of cot(2x)(cot(2x)+3csc(2x))

解答

\int cot (2 x) (cot (2 x) + 3 csc (2 x)) d x

- x - \frac{1}{2} cot (2 x) + \frac{3}{4} (- cot (x) - tan (x)) + C

求解步骤

\int cot (2 x) (cot (2 x) + 3 csc (2 x)) d x

使用三角恒等式改写

= \int \frac{cos ( 2 x ) ( cos ( 2 x ) + 3 )}{sin ^{2} ( 2 x )} d x

乘开

\frac{cos ( 2 x ) ( cos ( 2 x ) + 3 )}{sin ^{2} ( 2 x )} : \frac{cos ^{2} ( 2 x )}{sin ^{2} ( 2 x )} + \frac{3 cos ( 2 x )}{sin ^{2} ( 2 x )}

= \int \frac{cos ^{2} ( 2 x )}{sin ^{2} ( 2 x )} + \frac{3 cos ( 2 x )}{sin ^{2} ( 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 \frac{cos ^{2} ( 2 x )}{sin ^{2} ( 2 x )} d x + \int \frac{3 cos ( 2 x )}{sin ^{2} ( 2 x )} d x

\int \frac{cos ^{2} ( 2 x )}{sin ^{2} ( 2 x )} d x = - x - \frac{1}{2} cot (2 x)

\int \frac{3 cos ( 2 x )}{sin ^{2} ( 2 x )} d x = \frac{3}{4} (- cot (x) - tan (x))

= - x - \frac{1}{2} cot (2 x) + \frac{3}{4} (- cot (x) - tan (x))

解答补常数

= - x - \frac{1}{2} cot (2 x) + \frac{3}{4} (- cot (x) - tan (x)) + C

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

\int \frac{e ^{- 2 t}}{( 1 + e ^{- t} )} d t

\int \frac{2 ^{l n (x^{3})}}{16 x} d x

\int (5 x + 7)^{4} d x

\int e^{0.03 x} d x