integral of x(1-x^2)^{1/2}

Lösung

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

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

Schritte zur Lösung

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

Wende die partielle Integration an

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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