integral from-1 to 1 of xsin((npix)/2)

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int_{- \frac{πn}{2}}^{\frac{πn}{2}} \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_{- \frac{πn}{2}}^{\frac{πn}{2}} u sin (u) d u

Wende die partielle Integration an

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

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

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

Vereinfache

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

Berechne die Grenzen:

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

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

Vereinfache

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

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