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Beliebt Rechnen >

integral of xcos((pix)/2)

Lösung

\int x cos (\frac{π x}{2}) d x

Lösung

\frac{4}{π ^{2}} (\frac{π}{2} x sin (\frac{π x}{2}) + cos (\frac{π x}{2})) + C

Schritte zur Lösung

\int x cos (\frac{π x}{2}) d x

Wende U-Substitution an

= \int \frac{4 u cos ( u )}{π ^{2}} d u

Entferne die Konstante:

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

= \frac{4}{π ^{2}} \cdot \int u cos (u) d u

Wende die partielle Integration an

= \frac{4}{π ^{2}} (u sin (u) - \int sin (u) d u)

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

= \frac{4}{π ^{2}} (u sin (u) - (- cos (u)))

Setze in

u = \frac{π x}{2}

ein

= \frac{4}{π ^{2}} (\frac{π x}{2} sin (\frac{π x}{2}) - (- cos (\frac{π x}{2})))

Vereinfache

\frac{4}{π ^{2}} (\frac{π x}{2} sin (\frac{π x}{2}) - (- cos (\frac{π x}{2}))) : \frac{4}{π ^{2}} (\frac{π}{2} x sin (\frac{π x}{2}) + cos (\frac{π x}{2}))

= \frac{4}{π ^{2}} (\frac{π}{2} x sin (\frac{π x}{2}) + cos (\frac{π x}{2}))

Füge eine Konstante zur Lösung hinzu

= \frac{4}{π ^{2}} (\frac{π}{2} x sin (\frac{π x}{2}) + cos (\frac{π x}{2})) + C

Graph

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