integral from 0 to a/2 of xsin((npix)/a)

Lösung

\int_{0}^{\frac{a}{2}} x sin (\frac{nπ x}{a}) d x

\frac{a ^{2} sin ( \frac{πn}{2} )}{π ^{2} n ^{2}} - \frac{a ^{2} cos ( \frac{πn}{2} )}{2 πn}

Schritte zur Lösung

\int_{0}^{\frac{a}{2}} x sin (\frac{nπ x}{a}) d x

Wende die partielle Integration an

= [- a \frac{1}{πn} x cos (\frac{πn x}{a}) - \int - a \frac{1}{πn} cos (\frac{πn x}{a}) d x]_{0}^{\frac{a}{2}}

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

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

Vereinfache

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

Berechne die Grenzen:

\frac{a ^{2}}{π ^{2} n ^{2}} sin (\frac{πn}{2}) - \frac{a ^{2}}{2 πn} cos (\frac{πn}{2})

= \frac{a ^{2}}{π ^{2} n ^{2}} sin (\frac{πn}{2}) - \frac{a ^{2}}{2 πn} cos (\frac{πn}{2})

Vereinfache

= \frac{a ^{2} sin ( \frac{πn}{2} )}{π ^{2} n ^{2}} - \frac{a ^{2} cos ( \frac{πn}{2} )}{2 πn}

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