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Beliebt Rechnen >

integral of 1/2 (x-1/2 sin(2x))

Lösung

\int \frac{1}{2} (x - \frac{1}{2} sin (2 x)) d x

Lösung

\frac{1}{2} (\frac{x ^{2}}{2} + \frac{1}{4} cos (2 x)) + C

Schritte zur Lösung

\int \frac{1}{2} (x - \frac{1}{2} sin (2 x)) d x

Entferne die Konstante:

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

= \frac{1}{2} \cdot \int x - \frac{1}{2} sin (2 x) d x

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int x d x - \int \frac{1}{2} sin (2 x) d x)

\int x d x = \frac{x ^{2}}{2}

\int \frac{1}{2} sin (2 x) d x = - \frac{1}{4} cos (2 x)

= \frac{1}{2} (\frac{x ^{2}}{2} - (- \frac{1}{4} cos (2 x)))

Vereinfache

= \frac{1}{2} (\frac{x ^{2}}{2} + \frac{1}{4} cos (2 x))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (\frac{x ^{2}}{2} + \frac{1}{4} cos (2 x)) + C

Graph

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