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Beliebt Rechnen >

integral from 0 to 1 of xcos((npix)/2)

Lösung

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

Lösung

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

= \frac{1}{π ^{2} n ^{2}} \cdot \int_{0}^{πn} u cos (\frac{u}{2}) d u

Wende U-Substitution an

= \frac{1}{π ^{2} n ^{2}} \cdot \int_{0}^{\frac{πn}{2}} 4 v cos (v) d v

Entferne die Konstante:

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

= \frac{1}{π ^{2} n ^{2}} \cdot 4 \cdot \int_{0}^{\frac{πn}{2}} v cos (v) d v

Wende die partielle Integration an

= \frac{1}{π ^{2} n ^{2}} \cdot 4 [v sin (v) - \int sin (v) d v]_{0}^{\frac{πn}{2}}

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

= \frac{1}{π ^{2} n ^{2}} \cdot 4 [v sin (v) - (- cos (v))]_{0}^{\frac{πn}{2}}

Vereinfache

\frac{1}{π ^{2} n ^{2}} \cdot 4 [v sin (v) - (- cos (v))]_{0}^{\frac{πn}{2}} : \frac{4}{π ^{2} n ^{2}} [v sin (v) + cos (v)]_{0}^{\frac{πn}{2}}

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

Berechne die Grenzen:

\frac{πn}{2} sin (\frac{πn}{2}) + cos (\frac{πn}{2}) - 1

= \frac{4}{π ^{2} n ^{2}} (\frac{πn}{2} sin (\frac{πn}{2}) + cos (\frac{πn}{2}) - 1)

Vereinfache

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

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