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

Lösung

integral

x (a^{2} - x^{2})^{\frac{1}{2}}

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

Schritte zur Lösung

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

Wende U-Substitution an

= \int - \frac{u ^{\frac{1}{2}}}{2} d u

Entferne die Konstante:

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

= - \frac{1}{2} \cdot \int u^{\frac{1}{2}} d u

Wende die Potenzregel an

= - \frac{1}{2} \cdot \frac{2}{3} u^{\frac{3}{2}}

Setze in

u = a^{2} - x^{2}

ein

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

Vereinfache

- \frac{1}{2} \cdot \frac{2}{3} (a^{2} - x^{2})^{\frac{3}{2}} : - \frac{1}{3} (a^{2} - x^{2})^{\frac{3}{2}}

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

Füge eine Konstante zur Lösung hinzu

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

in t e g r a l \frac{1}{( x ( x ^{4} + 1 ) )}

in t e g r a l x^{9}

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in t e g r a l \frac{x ^{6}}{( x ^{3} + 1 )}

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