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

Lösung

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

\frac{1}{3} x^{3} sin^{2} (\frac{x}{2}) - \frac{1}{6} (- x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x)))) + C

Schritte zur Lösung

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

Wende die partielle Integration an

= \frac{1}{3} x^{3} sin^{2} (\frac{x}{2}) - \int \frac{1}{6} x^{3} sin (x) d x

\int \frac{1}{6} x^{3} sin (x) d x = \frac{1}{6} (- x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x))))

= \frac{1}{3} x^{3} sin^{2} (\frac{x}{2}) - \frac{1}{6} (- x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x))))

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} x^{3} sin^{2} (\frac{x}{2}) - \frac{1}{6} (- x^{3} cos (x) + 3 (x^{2} sin (x) - 2 (- x cos (x) + sin (x)))) + C

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