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Beliebt Rechnen >

integral of (xsin^2(x))/2

Lösung

\int \frac{x sin ^{2} ( x )}{2} d x

Lösung

\frac{1}{16} (2 x^{2} - 2 x sin (2 x) - cos (2 x)) + C

Schritte zur Lösung

\int \frac{x sin ^{2} ( x )}{2} 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 sin^{2} (x) d x

Wende die partielle Integration an

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

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

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

Vereinfache

\frac{1}{2} (\frac{1}{2} x (x - \frac{1}{2} sin (2 x)) - \frac{1}{2} (\frac{x ^{2}}{2} + \frac{1}{4} cos (2 x))) : \frac{1}{16} (2 x^{2} - 2 x sin (2 x) - cos (2 x))

= \frac{1}{16} (2 x^{2} - 2 x sin (2 x) - cos (2 x))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{16} (2 x^{2} - 2 x sin (2 x) - cos (2 x)) + C

Graph

Beliebte Beispiele

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\int \frac{8 + 4 x}{49 + 64 x ^{2}} d x

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\int \frac{3 ^{x - 1}}{sec ( 3 ^{x} )} d x

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\int sin (6 x) sin (8 x) d x

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\int \frac{x}{( x ^{2} + 5 ) ^{\frac{7}{2}}} d x

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\int x (5 - x^{2}) d x