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

Lösung

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

\frac{2}{27} (1 + 9 x)^{\frac{3}{2}} + C

Schritte zur Lösung

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

Vereinfache

1 + (3 x^{\frac{1}{2}})^{2} : 1 + 9 x

= \int 1 + 9 x d x

Wende U-Substitution an

= \int \frac{u}{9} d u

Entferne die Konstante:

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

= \frac{1}{9} \cdot \int u d u

Wende die Potenzregel an

= \frac{1}{9} \cdot \frac{2}{3} u^{\frac{3}{2}}

Setze in

u = 1 + 9 x

ein

= \frac{1}{9} \cdot \frac{2}{3} (1 + 9 x)^{\frac{3}{2}}

Vereinfache

\frac{1}{9} \cdot \frac{2}{3} (1 + 9 x)^{\frac{3}{2}} : \frac{2}{27} (1 + 9 x)^{\frac{3}{2}}

= \frac{2}{27} (1 + 9 x)^{\frac{3}{2}}

Füge eine Konstante zur Lösung hinzu

= \frac{2}{27} (1 + 9 x)^{\frac{3}{2}} + C

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