integral of (x-1)/(sqrt(2x+1))

Lösung

\int \frac{x - 1}{2 x + 1} d x

\frac{1}{3} x 2 x + 1 - \frac{4}{3} 2 x + 1 + C

Schritte zur Lösung

\int \frac{x - 1}{2 x + 1} d x

Multipliziere aus

\frac{x - 1}{2 x + 1} : \frac{x}{2 x + 1} - \frac{1}{2 x + 1}

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

Wende die Summenregel an:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int \frac{1}{2 x + 1} d x = 2 x + 1

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

Vereinfache

\frac{1}{3} (x - 1) 2 x + 1 - 2 x + 1 : \frac{1}{3} x 2 x + 1 - \frac{4}{3} 2 x + 1

= \frac{1}{3} x 2 x + 1 - \frac{4}{3} 2 x + 1

Füge eine Konstante zur Lösung hinzu

= \frac{1}{3} x 2 x + 1 - \frac{4}{3} 2 x + 1 + C

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