integral of x^3sqrt(a^2+x^2)

Lösung

\int x^{3} a^{2} + x^{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} d x

Vereinfache

x^{3} : x^{2} x

= \int x^{2} x a^{2} + x^{2} d x

Wende U-Substitution an

= \int \frac{u a ^{2} + u}{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 d u

Wende U-Substitution an

= \frac{1}{2} \cdot \int (v - a^{2}) v d v

Multipliziere aus

(v - a^{2}) v : v^{\frac{3}{2}} - a^{2} v

= \frac{1}{2} \cdot \int v^{\frac{3}{2}} - a^{2} v d v

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 v^{\frac{3}{2}} d v - \int a^{2} v d v)

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

\int a^{2} v d v = a^{2} \frac{2}{3} v^{\frac{3}{2}}

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

Ersetze zurück

= \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