integral from 0 to 2 of xsin((npi)/2 x)

Lösung

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

- \frac{4 ( - 1 ) ^{n}}{πn}

Schritte zur Lösung

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

\frac{nπ}{2} = nπ \frac{1}{2}

= \int_{0}^{2} x sin (nπ \frac{1}{2} x) d x

Wende U-Substitution an

= \int_{0}^{πn} \frac{4 u sin ( u )}{π ^{2} n ^{2}} d u

Entferne die Konstante:

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

= \frac{4}{π ^{2} n ^{2}} \cdot \int_{0}^{πn} u sin (u) d u

Wende die partielle Integration an

= \frac{4}{π ^{2} n ^{2}} [- u cos (u) - \int - cos (u) d u]_{0}^{πn}

\int - cos (u) d u = - sin (u)

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

Vereinfache

= \frac{4}{π ^{2} n ^{2}} [- u cos (u) + sin (u)]_{0}^{πn}

Berechne die Grenzen:

- π (- 1)^{n} n

= \frac{4}{π ^{2} n ^{2}} (- π (- 1)^{n} n)

Vereinfache

= - \frac{4 ( - 1 ) ^{n}}{πn}

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