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Beliebt Rechnen >

integral of ((x-2))/(sqrt(4x-x^2))

Lösung

\int \frac{( x - 2 )}{4 x - x ^{2}} d x

Lösung

- - x^{2} + 4 x + C

Schritte zur Lösung

\int \frac{x - 2}{4 x - x ^{2}} d x

Multipliziere aus

\frac{x - 2}{4 x - x ^{2}} : \frac{x}{4 x - x ^{2}} - \frac{2}{4 x - x ^{2}}

= \int \frac{x}{4 x - x ^{2}} - \frac{2}{4 x - x ^{2}} 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}{4 x - x ^{2}} d x - \int \frac{2}{4 x - x ^{2}} d x

\int \frac{x}{4 x - x ^{2}} d x = - - x^{2} + 4 x + 2 arcsin (\frac{1}{2} x - 1)

\int \frac{2}{4 x - x ^{2}} d x = 2 arcsin (\frac{1}{2} x - 1)

= - - x^{2} + 4 x + 2 arcsin (\frac{1}{2} x - 1) - 2 arcsin (\frac{1}{2} x - 1)

Vereinfache

= - - x^{2} + 4 x

Füge eine Konstante zur Lösung hinzu

= - - x^{2} + 4 x + C

Graph

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