integral of (1-cos(x))^3

解

\int (1 - cos (x))^{3} d x

\frac{5}{2} x + \frac{3}{4} sin (2 x) - 4 sin (x) + \frac{1}{3} sin^{3} (x) + C

解答ステップ

\int (1 - cos (x))^{3} d x

拡張

(1 - cos (x))^{3} : 1 - 3 cos (x) + 3 cos^{2} (x) - cos^{3} (x)

= \int 1 - 3 cos (x) + 3 cos^{2} (x) - cos^{3} (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 3 cos (x) d x + \int 3 cos^{2} (x) d x - \int cos^{3} (x) d x

\int 1 d x = x

\int 3 cos (x) d x = 3 sin (x)

\int 3 cos^{2} (x) d x = \frac{3}{2} (x + \frac{1}{2} sin (2 x))

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

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

簡素化

x - 3 sin (x) + \frac{3}{2} (x + \frac{1}{2} sin (2 x)) - (sin (x) - \frac{sin ^{3} ( x )}{3}) : \frac{5}{2} x + \frac{3}{4} sin (2 x) - 4 sin (x) + \frac{1}{3} sin^{3} (x)

= \frac{5}{2} x + \frac{3}{4} sin (2 x) - 4 sin (x) + \frac{1}{3} sin^{3} (x)

定数を解答に追加する

= \frac{5}{2} x + \frac{3}{4} sin (2 x) - 4 sin (x) + \frac{1}{3} sin^{3} (x) + C