integral from-1 to 1 of-xsin(npix)

Lösung

\int_{- 1}^{1} - x sin (nπ x) d x

\frac{2 ( - 1 ) ^{n}}{πn}

Schritte zur Lösung

\int_{- 1}^{1} - x sin (nπ x) d x

Entferne die Konstante:

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

= - \int_{- 1}^{1} x sin (nπ x) d x

Wende die partielle Integration an

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

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

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

Vereinfache

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

Berechne die Grenzen:

- (- 1)^{n} \frac{2}{πn}

= - (- (- 1)^{n} \frac{2}{πn})

Vereinfache

= \frac{2 ( - 1 ) ^{n}}{πn}

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