integral y(1-y^2)^{1/2}

Lösung

integral

y (1 - y^{2})^{\frac{1}{2}}

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

Schritte zur Lösung

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

Wende die partielle Integration an

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

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

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

Vereinfache

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

Füge eine Konstante zur Lösung hinzu

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

in t e g r a l 2 e^{- x}

in t e g r a l \frac{1}{( 1 + x ^{2} ) ^{0.5}}

in t e g r a l \frac{- 1}{y + 1}

in t e g r a l x^{2} e^{2} x

in t e g r a l \frac{2 x}{( x + y ) ^{3}}