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

Lösung

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

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

Schritte zur Lösung

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

Wende die partielle Integration an

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

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

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

Vereinfache

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

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

Füge eine Konstante zur Lösung hinzu

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

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