integral from-pi to pi of (x+pi)sin(nx)

Lösung

\int_{- π}^{π} (x + π) sin (n x) d x

- \frac{2 π ( - 1 ) ^{n}}{n}

Schritte zur Lösung

\int_{- π}^{π} (x + π) sin (n x) d x

Multipliziere aus

(x + π) sin (n x) : x sin (n x) + π sin (n x)

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

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

\int_{- π}^{π} π sin (n x) d x = 0

= - \frac{2 π ( - 1 ) ^{n}}{n} + 0

Vereinfache

= - \frac{2 π ( - 1 ) ^{n}}{n}

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