integral of cot(x)(-8sin(x)-10csc(x))

解

\int cot (x) (- 8 sin (x) - 10 csc (x)) d x

- 8 sin (x) + 10 csc (x) + C

解答ステップ

\int cot (x) (- 8 sin (x) - 10 csc (x)) d x

三角関数の公式を使用して書き換える

= \int \frac{cos ( x ) ( - 8 sin ^{2} ( x ) - 10 )}{sin ^{2} ( x )} d x

拡張

\frac{cos ( x ) ( - 8 sin ^{2} ( x ) - 10 )}{sin ^{2} ( x )} : - 8 cos (x) - \frac{10 cos ( x )}{sin ^{2} ( x )}

= \int - 8 cos (x) - \frac{10 cos ( x )}{sin ^{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 8 cos (x) d x - \int \frac{10 cos ( x )}{sin ^{2} ( x )} d x

\int 8 cos (x) d x = 8 sin (x)

\int \frac{10 cos ( x )}{sin ^{2} ( x )} d x = - 10 csc (x)

= - 8 sin (x) - (- 10 csc (x))

簡素化

= - 8 sin (x) + 10 csc (x)

定数を解答に追加する

= - 8 sin (x) + 10 csc (x) + C