integral from 0 to pi of xcos((nx)/2)

Lösung

\int_{0}^{π} x cos (\frac{n x}{2}) d x

\frac{2 πn sin ( \frac{πn}{2} ) + 4 cos ( \frac{πn}{2} ) - 4}{n ^{2}}

Schritte zur Lösung

\int_{0}^{π} x cos (\frac{n x}{2}) d x

Wende die partielle Integration an

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

\int \frac{2}{n} sin (\frac{n x}{2}) d x = - \frac{4}{n ^{2}} cos (\frac{n x}{2})

= [\frac{1}{n} (2 x sin (\frac{n x}{2}) - n (- \frac{4}{n ^{2}} cos (\frac{n x}{2})))]_{0}^{π}

Vereinfache

[\frac{1}{n} (2 x sin (\frac{n x}{2}) - n (- \frac{4}{n ^{2}} cos (\frac{n x}{2})))]_{0}^{π} : [\frac{1}{n} (2 x sin (\frac{n x}{2}) + \frac{4}{n} cos (\frac{n x}{2}))]_{0}^{π}

= [\frac{1}{n} (2 x sin (\frac{n x}{2}) + \frac{4}{n} cos (\frac{n x}{2}))]_{0}^{π}

Berechne die Grenzen:

\frac{1}{n} (2 π sin (\frac{nπ}{2}) + \frac{4}{n} cos (\frac{nπ}{2})) - \frac{4}{n ^{2}}

= \frac{1}{n} (2 π sin (\frac{nπ}{2}) + \frac{4}{n} cos (\frac{nπ}{2})) - \frac{4}{n ^{2}}

Vereinfache

= \frac{2 πn sin ( \frac{πn}{2} ) + 4 cos ( \frac{πn}{2} ) - 4}{n ^{2}}

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