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Beliebt Rechnen >

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

Lösung

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

Lösung

\frac{1}{5} (1 + x^{2})^{\frac{5}{2}} + C

Schritte zur Lösung

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

Wende U-Substitution an

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

Wende die Potenzregel an

= \frac{1}{2} \cdot \frac{2}{5} u^{\frac{5}{2}}

Setze in

u = 1 + x^{2}

ein

= \frac{1}{2} \cdot \frac{2}{5} (1 + x^{2})^{\frac{5}{2}}

Vereinfache

\frac{1}{2} \cdot \frac{2}{5} (1 + x^{2})^{\frac{5}{2}} : \frac{1}{5} (1 + x^{2})^{\frac{5}{2}}

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

Füge eine Konstante zur Lösung hinzu

= \frac{1}{5} (1 + x^{2})^{\frac{5}{2}} + C

Graph

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