integral of sin(x)(2+cos(x))^2

解

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

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

解答ステップ

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

拡張

sin (x) (2 + cos (x))^{2} : 4 sin (x) + 4 sin (x) cos (x) + cos^{2} (x) sin (x)

= \int 4 sin (x) + 4 sin (x) cos (x) + cos^{2} (x) sin (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 4 sin (x) d x + \int 4 sin (x) cos (x) d x + \int cos^{2} (x) sin (x) d x

\int 4 sin (x) d x = - 4 cos (x)

\int 4 sin (x) cos (x) d x = 2 sin^{2} (x)

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

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

定数を解答に追加する

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