integral of x^3(a^2-x^2)^{1/2}

Lösung

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

- \frac{1}{15} (3 x^{2} + 2 a^{2}) (- x^{2} + a^{2})^{\frac{3}{2}} + C

Schritte zur Lösung

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

Wende U-Substitution an

= \int - \frac{( - u + a ^{2} ) 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 + a^{2}) u^{\frac{1}{2}} d u

Multipliziere aus

(- u + a^{2}) u^{\frac{1}{2}} : - u^{\frac{3}{2}} + a^{2} u^{\frac{1}{2}}

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

= - \frac{1}{2} (- \int u^{\frac{3}{2}} d u + \int a^{2} u^{\frac{1}{2}} d u)

\int u^{\frac{3}{2}} d u = \frac{2}{5} u^{\frac{5}{2}}

\int a^{2} u^{\frac{1}{2}} d u = a^{2} \frac{2}{3} u^{\frac{3}{2}}

= - \frac{1}{2} (- \frac{2}{5} u^{\frac{5}{2}} + a^{2} \frac{2}{3} u^{\frac{3}{2}})

Setze in

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

ein

= - \frac{1}{2} (- \frac{2}{5} (a^{2} - x^{2})^{\frac{5}{2}} + a^{2} \frac{2}{3} (a^{2} - x^{2})^{\frac{3}{2}})

Vereinfache

- \frac{1}{2} (- \frac{2}{5} (a^{2} - x^{2})^{\frac{5}{2}} + a^{2} \frac{2}{3} (a^{2} - x^{2})^{\frac{3}{2}}) : - \frac{1}{15} (3 x^{2} + 2 a^{2}) (- x^{2} + a^{2})^{\frac{3}{2}}

= - \frac{1}{15} (3 x^{2} + 2 a^{2}) (- x^{2} + a^{2})^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= - \frac{1}{15} (3 x^{2} + 2 a^{2}) (- x^{2} + a^{2})^{\frac{3}{2}} + C

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