integral from 0 to pi of t^2sin(nt)

Lösung

\int_{0}^{π} t^{2} sin (n t) d t

\frac{2 ( - 1 ) ^{n} - π ^{2} ( - 1 ) ^{n} n ^{2} - 2}{n ^{3}}

Schritte zur Lösung

\int_{0}^{π} t^{2} sin (n t) d t

Wende die partielle Integration an

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

\int - \frac{2}{n} t cos (n t) d t = - \frac{2}{n ^{3}} (n t sin (n t) + cos (n t))

= [\frac{1}{n} (- t^{2} cos (n t) - n (- \frac{2}{n ^{3}} (n t sin (n t) + cos (n t))))]_{0}^{π}

Vereinfache

[\frac{1}{n} (- t^{2} cos (n t) - n (- \frac{2}{n ^{3}} (n t sin (n t) + cos (n t))))]_{0}^{π} : [\frac{1}{n} (- t^{2} cos (n t) + \frac{2}{n ^{2}} (n t sin (n t) + cos (n t)))]_{0}^{π}

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

Berechne die Grenzen:

\frac{1}{n} ((- 1)^{n} \frac{2}{n ^{2}} - π^{2} (- 1)^{n}) - \frac{2}{n ^{3}}

= \frac{1}{n} ((- 1)^{n} \frac{2}{n ^{2}} - π^{2} (- 1)^{n}) - \frac{2}{n ^{3}}

Vereinfache

= \frac{2 ( - 1 ) ^{n} - π ^{2} ( - 1 ) ^{n} n ^{2} - 2}{n ^{3}}

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