integral of x(xsin(x)-cos(x))

解

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

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

解答ステップ

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

拡張

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

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

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

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

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

簡素化

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

定数を解答に追加する

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