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Beliebt Rechnen >

integral of (x^2-2x+5)/(sqrt(9-x^2))

Lösung

\int \frac{x ^{2} - 2 x + 5}{9 - x ^{2}} d x

Lösung

\frac{1}{2} (9 arcsin (\frac{x}{3}) - x 9 - x^{2}) + 2 9 - x^{2} + 5 arcsin (\frac{1}{3} x) + C

Schritte zur Lösung

\int \frac{x ^{2} - 2 x + 5}{9 - x ^{2}} d x

Multipliziere aus

\frac{x ^{2} - 2 x + 5}{9 - x ^{2}} : \frac{x ^{2}}{9 - x ^{2}} - \frac{2 x}{9 - x ^{2}} + \frac{5}{9 - x ^{2}}

= \int \frac{x ^{2}}{9 - x ^{2}} - \frac{2 x}{9 - x ^{2}} + \frac{5}{9 - 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 ^{2}}{9 - x ^{2}} d x - \int \frac{2 x}{9 - x ^{2}} d x + \int \frac{5}{9 - x ^{2}} d x

\int \frac{x ^{2}}{9 - x ^{2}} d x = \frac{1}{2} (9 arcsin (\frac{x}{3}) - x 9 - x^{2})

\int \frac{2 x}{9 - x ^{2}} d x = - 2 9 - x^{2}

\int \frac{5}{9 - x ^{2}} d x = 5 arcsin (\frac{1}{3} x)

= \frac{1}{2} (9 arcsin (\frac{x}{3}) - x 9 - x^{2}) - (- 2 9 - x^{2}) + 5 arcsin (\frac{1}{3} x)

Vereinfache

= \frac{1}{2} (9 arcsin (\frac{x}{3}) - x 9 - x^{2}) + 2 9 - x^{2} + 5 arcsin (\frac{1}{3} x)

Füge eine Konstante zur Lösung hinzu

= \frac{1}{2} (9 arcsin (\frac{x}{3}) - x 9 - x^{2}) + 2 9 - x^{2} + 5 arcsin (\frac{1}{3} x) + C

Graph

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