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

Lösung

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

- - x^{2} + 2 x + C

Schritte zur Lösung

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

Multipliziere aus

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

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

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

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

= - - x^{2} + 2 x + arcsin (x - 1) - arcsin (x - 1)

Vereinfache

= - - x^{2} + 2 x

Füge eine Konstante zur Lösung hinzu

= - - x^{2} + 2 x + C

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