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Beliebt Rechnen >

integral of xcos((npix)/4)

Lösung

\int x cos (\frac{nπ x}{4}) d x

Lösung

\frac{1}{π ^{2} n ^{2}} (4 πn x sin (\frac{πn x}{4}) + 16 cos (\frac{πn x}{4})) + C

Schritte zur Lösung

\int x cos (\frac{nπ x}{4}) d x

Wende U-Substitution an

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

Wende die partielle Integration an

= \frac{1}{π ^{2} n ^{2}} (4 u sin (\frac{u}{4}) - \int 4 sin (\frac{u}{4}) d u)

\int 4 sin (\frac{u}{4}) d u = - 16 cos (\frac{u}{4})

= \frac{1}{π ^{2} n ^{2}} (4 u sin (\frac{u}{4}) - (- 16 cos (\frac{u}{4})))

Setze in

u = nπ x

ein

= \frac{1}{π ^{2} n ^{2}} (4 nπ x sin (\frac{nπ x}{4}) - (- 16 cos (\frac{nπ x}{4})))

Vereinfache

= \frac{1}{π ^{2} n ^{2}} (4 πn x sin (\frac{πn x}{4}) + 16 cos (\frac{πn x}{4}))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{π ^{2} n ^{2}} (4 πn x sin (\frac{πn x}{4}) + 16 cos (\frac{πn x}{4})) + C

Graph

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