integral of (1-cos(x))/(csc^2(x))

解

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

\frac{1}{2} (x - \frac{1}{2} sin (2 x)) - \frac{sin ^{3} ( x )}{3} + C

解答ステップ

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

拡張

\frac{1 - cos ( x )}{csc ^{2} ( x )} : \frac{1}{csc ^{2} ( x )} - \frac{cos ( x )}{csc ^{2} ( x )}

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

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

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

= \frac{1}{2} (x - \frac{1}{2} sin (2 x)) - \frac{sin ^{3} ( x )}{3}

定数を解答に追加する

= \frac{1}{2} (x - \frac{1}{2} sin (2 x)) - \frac{sin ^{3} ( x )}{3} + C