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Beliebt Rechnen >

integral of (1-x)/(sqrt(5-2x-x^2))

Lösung

\int \frac{1 - x}{5 - 2 x - x ^{2}} d x

Lösung

2 arcsin (\frac{1}{6} (x + 1)) + - x^{2} - 2 x + 5 + C

Schritte zur Lösung

\int \frac{1 - x}{5 - 2 x - x ^{2}} d x

Multipliziere aus

\frac{1 - x}{5 - 2 x - x ^{2}} : \frac{1}{5 - 2 x - x ^{2}} - \frac{x}{5 - 2 x - x ^{2}}

= \int \frac{1}{5 - 2 x - x ^{2}} - \frac{x}{5 - 2 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{1}{5 - 2 x - x ^{2}} d x - \int \frac{x}{5 - 2 x - x ^{2}} d x

\int \frac{1}{5 - 2 x - x ^{2}} d x = arcsin (\frac{1}{6} (x + 1))

\int \frac{x}{5 - 2 x - x ^{2}} d x = - - x^{2} - 2 x + 5 - arcsin (\frac{1}{6} (x + 1))

= arcsin (\frac{1}{6} (x + 1)) - (- - x^{2} - 2 x + 5 - arcsin (\frac{1}{6} (x + 1)))

Vereinfache

= 2 arcsin (\frac{1}{6} (x + 1)) + - x^{2} - 2 x + 5

Füge eine Konstante zur Lösung hinzu

= 2 arcsin (\frac{1}{6} (x + 1)) + - x^{2} - 2 x + 5 + C

Graph

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