integral of y^3sqrt(3-y^2)

Lösung

\int y^{3} 3 - y^{2} d y

\frac{1}{5} (3 - y^{2})^{\frac{5}{2}} - (3 - y^{2})^{\frac{3}{2}} + C

Schritte zur Lösung

\int y^{3} 3 - y^{2} d y

Wende U-Substitution an

= \int - (- u^{2} + 3) u^{2} d u

Entferne die Konstante:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int (- u^{2} + 3) u^{2} d u

Multipliziere aus

(- u^{2} + 3) u^{2} : - u^{4} + 3 u^{2}

= - \int - u^{4} + 3 u^{2} d u

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - (- \int u^{4} d u + \int 3 u^{2} d u)

\int u^{4} d u = \frac{u ^{5}}{5}

\int 3 u^{2} d u = u^{3}

= - (- \frac{u ^{5}}{5} + u^{3})

Setze in

u = 3 - y^{2}

ein

= - - \frac{( 3 - y ^{2} ) ^{5}}{5} + (3 - y^{2})^{3}

Vereinfache

- - \frac{( 3 - y ^{2} ) ^{5}}{5} + (3 - y^{2})^{3} : \frac{1}{5} (3 - y^{2})^{\frac{5}{2}} - (3 - y^{2})^{\frac{3}{2}}

= \frac{1}{5} (3 - y^{2})^{\frac{5}{2}} - (3 - y^{2})^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{1}{5} (3 - y^{2})^{\frac{5}{2}} - (3 - y^{2})^{\frac{3}{2}} + C

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