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

Lösung

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

2 x x - 1 - \frac{x}{x - 1} x + \frac{x}{x - 1} - ln x + x - 1 + C

Schritte zur Lösung

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

Wende die partielle Integration an

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

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

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

Vereinfache

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

= 2 x x - 1 - \frac{x}{x - 1} x + \frac{x}{x - 1} - ln x + x - 1

Füge eine Konstante zur Lösung hinzu

= 2 x x - 1 - \frac{x}{x - 1} x + \frac{x}{x - 1} - ln x + x - 1 + C

\int 25.89 e^{- 0.993 x} + 6.39 d x

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