integral of csc(x)(sin(x)+cot(x))

解

\int csc (x) (sin (x) + cot (x)) d x

x - csc (x) + C

解答ステップ

\int csc (x) (sin (x) + cot (x)) d x

三角関数の公式を使用して書き換える

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

拡張

\frac{sin ^{2} ( x ) + cos ( x )}{sin ^{2} ( x )} : 1 + \frac{cos ( x )}{sin ^{2} ( x )}

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

\int 1 d x = x

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

= x - csc (x)

定数を解答に追加する

= x - csc (x) + C