integral from-1 to 1 of-t*e^{-iwt}

Lösung

\int_{- 1}^{1} - t \cdot e^{- i wt} d t

i \frac{2 sin ( w ) - 2 w cos ( w )}{w ^{2}}

Schritte zur Lösung

\int_{- 1}^{1} - t e^{- i wt} d t

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int_{- 1}^{1} e^{- w i t} t d t

Wende U-Substitution an

= - \int_{i w}^{- i w} - \frac{e ^{u} u}{w ^{2}} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - (- \frac{1}{w ^{2}} \cdot \int_{i w}^{- i w} e^{u} u d u)

Wende die partielle Integration an

= - (- \frac{1}{w ^{2}} [e^{u} u - \int e^{u} d u]_{i w}^{- i w})

\int e^{u} d u = e^{u}

= - (- \frac{1}{w ^{2}} [e^{u} u - e^{u}]_{i w}^{- i w})

Vereinfache

- (- \frac{1}{w ^{2}} [e^{u} u - e^{u}]_{i w}^{- i w}) : \frac{[ e ^{u} u - e ^{u} ] _{i w}^{- i w}}{w ^{2}}

= \frac{[ e ^{u} u - e ^{u} ] _{i w}^{- i w}}{w ^{2}}

Berechne die Grenzen:

i (2 sin (w) - 2 w cos (w))

= \frac{i ( 2 sin ( w ) - 2 w cos ( w ) )}{w ^{2}}

Vereinfache

= i \frac{2 sin ( w ) - 2 w cos ( w )}{w ^{2}}

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