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

解

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

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

解答ステップ

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

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

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

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

簡素化

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

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

定数を解答に追加する

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