integral from-pi to 0 of xsin(nx)

Lösung

\int_{- π}^{0} x sin (n x) d x

- \frac{π ( - 1 ) ^{n}}{n}

Schritte zur Lösung

\int_{- π}^{0} x sin (n x) d x

Wende die partielle Integration an

= [- \frac{1}{n} x cos (n x) - \int - \frac{1}{n} cos (n x) d x]_{- π}^{0}

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

= [- \frac{1}{n} x cos (n x) - (- \frac{1}{n ^{2}} sin (n x))]_{- π}^{0}

Vereinfache

= [- \frac{1}{n} x cos (n x) + \frac{1}{n ^{2}} sin (n x)]_{- π}^{0}

Berechne die Grenzen:

- (- 1)^{n} \frac{π}{n}

= - (- 1)^{n} \frac{π}{n}

Vereinfache

= - \frac{π ( - 1 ) ^{n}}{n}

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