integral of xe^{-inpix}

解

\int x e^{- inπ x} d x

\frac{1}{π ^{2} n ^{2}} (cos (πn x) + πn x sin (πn x)) + i \frac{1}{π ^{2} n ^{2}} (- sin (πn x) + πn x cos (πn x)) + C

解答ステップ

\int x e^{- inπ x} d x

簡素化

x e^{- inπ x} : x cos (πn x) - i x sin (πn x)

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

\int x cos (πn x) d x = \frac{1}{π ^{2} n ^{2}} (πn x sin (πn x) + cos (πn x))

\int i x sin (πn x) d x = i \frac{1}{π ^{2} n ^{2}} (sin (πn x) - πn x cos (πn x))

= \frac{1}{π ^{2} n ^{2}} (πn x sin (πn x) + cos (πn x)) - i \frac{1}{π ^{2} n ^{2}} (sin (πn x) - πn x cos (πn x))

簡素化

\frac{1}{π ^{2} n ^{2}} (πn x sin (πn x) + cos (πn x)) - i \frac{1}{π ^{2} n ^{2}} (sin (πn x) - πn x cos (πn x)) : \frac{1}{π ^{2} n ^{2}} (cos (πn x) + πn x sin (πn x)) + i \frac{1}{π ^{2} n ^{2}} (- sin (πn x) + πn x cos (πn x))

= \frac{1}{π ^{2} n ^{2}} (cos (πn x) + πn x sin (πn x)) + i \frac{1}{π ^{2} n ^{2}} (- sin (πn x) + πn x cos (πn x))

定数を解答に追加する

= \frac{1}{π ^{2} n ^{2}} (cos (πn x) + πn x sin (πn x)) + i \frac{1}{π ^{2} n ^{2}} (- sin (πn x) + πn x cos (πn x)) + C