integral of (sin(x)+cos(x)-x)

解

\int (sin (x) + cos (x) - x) d x

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

解答ステップ

\int sin (x) + cos (x) - 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 sin (x) d x + \int cos (x) d x - \int x d x

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

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

\int x d x = \frac{x ^{2}}{2}

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

定数を解答に追加する

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