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

解

\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

解答ステップ

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

三角関数の公式を使用して書き換える

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

拡張

\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

総和規則を適用する:

\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))

定数を解答に追加する

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