integral of (csc^2(4x))/(cot(4x))

解

\int \frac{csc ^{2} ( 4 x )}{cot ( 4 x )} d x

- \frac{1}{4} ln ∣ cos (4 x) ∣ + \frac{1}{4} ln ∣ sin (4 x) ∣ + C

解答ステップ

\int \frac{csc ^{2} ( 4 x )}{cot ( 4 x )} d x

三角関数の公式を使用して書き換える

= \int \frac{1 + cot ^{2} ( 4 x )}{cot ( 4 x )} d x

拡張

\frac{1 + cot ^{2} ( 4 x )}{cot ( 4 x )} : \frac{1}{cot ( 4 x )} + cot (4 x)

= \int \frac{1}{cot ( 4 x )} + cot (4 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{1}{cot ( 4 x )} d x + \int cot (4 x) d x

\int \frac{1}{cot ( 4 x )} d x = - \frac{1}{4} ln ∣ cos (4 x) ∣

\int cot (4 x) d x = \frac{1}{4} ln ∣ sin (4 x) ∣

= - \frac{1}{4} ln ∣ cos (4 x) ∣ + \frac{1}{4} ln ∣ sin (4 x) ∣

定数を解答に追加する

= - \frac{1}{4} ln ∣ cos (4 x) ∣ + \frac{1}{4} ln ∣ sin (4 x) ∣ + C