integral of xe^{ix}

解

\int x e^{i x} d x

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

解答ステップ

\int x e^{i x} d x

簡素化

x e^{i x} : x cos (x) + i x sin (x)

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

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

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

= x sin (x) + cos (x) + i (sin (x) - x cos (x))

定数を解答に追加する

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