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

Lösung

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

\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