integral of x^2sin((npix)/l)

Lösung

\int x^{2} sin (\frac{nπ x}{l}) d x

- l \frac{1}{πn} x^{2} cos (\frac{πn x}{l}) + \frac{2 l}{πn} (l \frac{1}{πn} x sin (\frac{πn x}{l}) + l \frac{l}{π ^{2} n ^{2}} cos (\frac{πn x}{l})) + C

Schritte zur Lösung

\int x^{2} sin (\frac{nπ x}{l}) d x

Wende die partielle Integration an

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

= - l \frac{1}{πn} x^{2} cos (\frac{πn x}{l}) + \frac{2 l}{πn} (l \frac{1}{πn} x sin (\frac{πn x}{l}) + l \frac{l}{π ^{2} n ^{2}} cos (\frac{πn x}{l})) + C

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