integral of cos(x)(6sin^2(x)-1)

解

\int cos (x) (6 sin^{2} (x) - 1) d x

- sin (x) cos (2 x) + C

解答ステップ

\int cos (x) (6 sin^{2} (x) - 1) d x

拡張

cos (x) (6 sin^{2} (x) - 1) : 6 sin^{2} (x) cos (x) - cos (x)

= \int 6 sin^{2} (x) cos (x) - cos (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 6 sin^{2} (x) cos (x) d x - \int cos (x) d x

\int 6 sin^{2} (x) cos (x) d x = 2 sin^{3} (x)

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

= 2 sin^{3} (x) - sin (x)

簡素化

2 sin^{3} (x) - sin (x) : - sin (x) cos (2 x)

= - sin (x) cos (2 x)

定数を解答に追加する

= - sin (x) cos (2 x) + C