integral of (ax+b)^{3/2}

Lösung

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

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

Schritte zur Lösung

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

Wende U-Substitution an

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

Entferne die Konstante:

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

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

Wende die Potenzregel an

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

Setze in

u = a x + b

ein

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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