integral from-pi to pi of x*e^{-inx}

Lösung

\int_{- π}^{π} x \cdot e^{- in x} d x

\frac{2 πi ( - 1 ) ^{n}}{n}

Schritte zur Lösung

\int_{- π}^{π} x e^{- in x} d x

Vereinfache

x e^{- in x} : x cos (n x) - i x sin (n x)

= \int_{- π}^{π} x cos (n x) - i x sin (n x) d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{- π}^{π} x cos (n x) d x - \int_{- π}^{π} i x sin (n x) d x

\int_{- π}^{π} x cos (n x) d x = 0

\int_{- π}^{π} i x sin (n x) d x = - \frac{2 πi ( - 1 ) ^{n}}{n}

= 0 - (- \frac{2 πi ( - 1 ) ^{n}}{n})

Vereinfache

= \frac{2 πi ( - 1 ) ^{n}}{n}

\int_{1}^{6} \frac{4}{x} - \frac{4}{x ^{2}} d x

\int_{1}^{e} x lo g_{e} (x) d x

\int_{0}^{7} 6 x \cdot 3 x + 1 d x

\int_{0}^{1} 3 y (1 - y)^{2} d y

\int_{0}^{1} (x + \frac{1}{2}) d x